Are there any examples of overpopulation?

Easter Island is one example. From the Wikipedia article:

It is believed that Easter Island's Polynesian inhabitants arrived on Easter Island sometime between 700 and 1100 CE. They created a thriving and industrious culture as evidenced by the island's numerous enormous stone moai and other artifacts. However, human activity, the introduction of the Polynesian rat and overpopulation led to gradual deforestation and extinction of many important natural resources, which severely weakened the Rapa Nui civilization. By the time of European arrival in 1722, the island's population had dropped to 2,000–3,000 from an estimated high of approximately 15,000 just a century earlier.

As noted in the discussion on the question, there are two kinds of overpopulation which I'll call "chronic" and "acute". Chronic overpopulation, where the full production capacity of agriculture in normal times is insufficient to feed the population, is sometimes the only case people consider. But acute overpopulation, where the population "only" starves in times of famine, is also over population, because famines caused by things like droughts or blights (or war or plagues of locusts) are an inescapable part of the landscape.

It seems to me that the fact of acute overpopulation makes chronic overpopulation rare -- before you get so many people that you strain food production in good times, you get a famine when times turn bad.

Given that, pretty much any major famine (one big enough to kill a significant number of people) is evidence of overpopulation. (The exception is man-made famines, e.g., the ones caused by Stalin in the 30s and Mao in the 50s.)

Maybe not

Joseph Tainter argues, in Archeology of Overshoot and Collapse, that so far there is no evidence of a Malthusian catastrophe. Whether you follow Tainter in this assertion will in part depend on how wide or narrow your definition of overpopulation is.

Problem with definition

According to Malthus, population grows exponentially while food production grows linearily, until the first outstrips the latter. This definition of overpoulation is IMO overly narrow, as any case where food production drops would be excluded. As we see, in numerous cases food production would rise for a while and then collapse through environmental degradation or other causes. This overshoot theory is applied in the case studies below.

Case studies

Tainter adresses several cases commonly cited as overshoot, I'll cite only conclusions regarding the examples from Jared Diamond's "Collapse":

Henderson and Pitcairn Islands:

Pitcairn and Henderson islands, for example, are small, remote, and lacking critical resources. They could not be occupied for long without obtaining resources from elsewhere. The failure of attempts to occupy them (which I do not consider collapses) is attributed by Diamond as much to problems plaguing trade partners on Mangareva as to anything done by the occupants of Pitcairn and Henderson islands (pp. 120–35). For comparison, if the resupply of the International Space Station were to fail, future historians would not wonder at the fate of the astronauts, nor draw broader inferences.

Norse Greenland:

Norse Greenland is not an enduring lesson for the same reason: Local resources could not indefinitely support a medieval European society. [...] the dilemma is not that the Greenland Norse went extinct or left, but that they need not have done either. The continued occupation of the area by the Inuit shows that alternative subsistence strategies and ways of life would have allowed the Norse to survive in Greenland.

This also points to a problem with defining overpopulation: The "carrying capacity" will always depend on agricultural and other technologies used, consumption patterns etc.

Chaco Canyon, Anasazi & southwestern Native Americans, Maya:

Diamond would have liked to have shown that Chaco Canyon, the Anasazi and other Southwesterners, and the Maya overshot the capacity of their environments, degraded them, and collapsed. In each case, though, he was confronted with the potential roles of climate and other factors. These were not Malthusian overshoots. In Diamond’s formulation, these cases may illustrate overshoot in reference to extreme climatic conditions. If the extreme conditions had not occurred, the societies in question might not have collapsed

Easter Island:
The case of Easter Island, deforestation is often seen as the primary cause of collapse:

Several authors see the deforestation as the start of a cascading process that led to a decline in fishing and farming; changes in farming technology; increases in warfare and insecurity; changes in settlement patterns; population decline; and sociopolitical collapse. Forest depletion, in this view, led to a shortage of wood for canoe construction, and thus to a decline in the consumption of fish (especially deep, pelagic fish) and marine mammals.

However, Tainter doubts the deforestation necessarily has these consequences:

Even if lack of wood meant a decline in the availability of sea mammals and large fish, the entire population would not have been equally affected. As is common in Polynesia, the high-ranking Miru clan controlled fishing in deeper, offshore waters. [...] If deforestation did lead to a decline in the catch of larger fish and marine mammals, a large part of the population was only minimally affected.

A decline in agriculture would have been more serious than the loss of marine delicacies. Easter Island undoubtedly experienced erosion. It is a common mistake, though, to assume that erosion is always detrimental. Ancient Egypt and Mesopotamia were sustained by upstream erosion, as other places have been. No research has shown that erosion adversely affected Easter Island agriculture. Removing forest cover would have exposed soils to drying and challenged young plantings. The islanders responded by digging pits, erecting small windbreaks, and employing lithic mulch. Problems of soil fertility could have been addressed by shifting cultivation and/or by use of night soil.

Nearly the entire surface of Easter Island is arable, yet in recent centuries only the coast and the interior of Rano Kau crater were intensively cultivated. The factor limiting agriculture may have been water, rather than erosion, wind, or soil fertility. If agricultural productivity declined with deforestation, this could have been compensated for by increasing the area under intensive cultivation. [...] When people face agricultural problems, a common response is to intensify production. This is what the Easter Islanders did

Tainter draws the conclusion that we cannot show that the Polynesians overextended the "carrying capacity" of their islands, and thus cannot prove that the Easter Islands are a case of ecological overshoot.

Conclusion
You can always argue that every case where another immediate cause (drought, war ...) can be identified as the culprit for the collapse of a society, this immediate cuplrit was the way Malthus's harsh law manifests itself. This is not entirely wrong - there are hard thermodynamic limits on how much life (as activity, not biomass) Earth can support - but in my eyes not really useful. The more useful question would be - and this is where I find Tainter more convincing than in his many quibbles with Diamond and Chew - how a society handles resources & consumption. Diamond (in Collapse) gives an, in my eyes, instructive example from the ultimately doomed Henderson & Pitcairn islands: Archeological evidence seems to show that the islanders imported pigs, until apparently all pigs were culled - it appears they saw that raising pigs yields less food than eating the pig fodder themselves, and adapted accordingly. So the question is/would be, what adaptions are possible today?

Edit to add - Further criticism of Malthus:
I came across a further criticism of Malthus's overshoot thesis that I will provide here as I belive it's relevant to the OPs question - TL,DR: Malthus's central assumption of exponential growth kept in check by hunger (or other available resources) does not match actual human behavior, though it may have seemed to him so at the time due to specific historic circumstances. Malthus's thesis was motivated as much by his politics as by data and reasoning:

Human population is stable whenever the birth rate equals the death rate. If the stabilisation of population is caused by famine, it would mean the death rate rises to match the birth rate. In fact, both birth and death rates fall. Before the advent of modern medicine, birth rates and death rates were high, towns were disease-ridden population sinks, and the population was therefore predominantly young and rural. With the advent of modern understandings of disease, a series of changes lead to falling death rates, falling birth rates, urbanisation, and an aging population. This is known as the demographic transition.

The seemingly exponential growth observed by Malthus was in fact the demographic transition between the high birth/death equilibrium to the low birth/death equilibrium. This transition seems to follow a more or less universal pattern, generating a chain of positive feedbacks once it begins. The most developed countries began this transition several centuries ago and are now mostly at the higher, older, urban equilibrium (population decline is even a concern in some places). Many less developed countries are not yet at the higher equilibrium, have younger, more rural populations, and are still experiencing rapid population growth. UN demographers expect the world population to stabilise somewhere in the 9 billion region.

Writing in 1798, Malthus mistook the rapid growth phase of a sigmoid curve for an exponential one. In fact, Malthus' main thrust was not to advance a theory of human ecology, but to make a political attack on the poor laws and the idea of raising workers' wages.

(libcom - the dangers of reactionary ecology)

Because people are quibbling in comments over the meaning of "overpopulation", it might suggest that this question is broad or unclear. So let's look at it from a different angle.

In the 1960's, Dr. John Calhoun, working at NIH, conducted his famous (some would say infamous) "mouse utopia experiment".

Mice were placed in an enclosure (named "Universe 25") where they were provided with an unlimited supply of food and water. The only constraint was the size of enclosure.

For a while, the mouse population exploded, as one might expect. After a while the population leveled off, and in the crowded conditions, several peculiar patterns of behavior emerged. For example, there were gangs of unattached male mice who would get into violent fights with each other, and individuals that Calhoun dubbed "the beautiful ones" who would do nothing but eat, groom, and sleep.

Eventually, all of the mice just went about their business, ignoring each other as if they were on their phones all the time, and the population plummeted. In fact, none of the mice survived despite having unlimited resources. The devolution of the mice's social behavior (called a "behavioral sink" by Calhoun), and only that, led to their extinction.

Other researchers produced similar results using rats. It isn't clear how well this experiment applies to human behavior, and Calhoun's methodology has been criticized, but it is a good example of the effects of overcrowding that we should think about.