Did the apostolic or early church fathers acknowledge Papal infallibility?

St. Robert Bellarmine, On the Roman Pontiff (from De Controversiis), vol. 2, Book IV: "On the Supreme Spiritual Power of the Pope", ch. I: "Whether the Pope Is the Supreme Judge in Controversies of Faith and Morals" quotes:

St. Jerome, a most learned man, [who] in the question on the three hypostases did not trust his own erudition, nor the opinion of the Eastern bishops, nor even in the authority of his own bishop, Paul, the Patriarch of Antioch. Rather he wrote to Pope Damasus [in c. 376 A.D.], saying:

I, a sheep, demand assistance from the pastor. Please, discern, for I will not fear to say three hypostases if you bid it so.

^{Epist., 15, n. 2 (ML, XXII, 355).}

Theodoret [✝457 A.D.], who was also very learned among the Greek Fathers, wrote to Pope Leo I, saying

If Paul, the herald of truth, the trumpet of the Holy Spirit, hastened to the great Peter to carry from him the solution to the difficulties of those at Antioch who hesitated from conformity with the law, much more do we, men insignificant and small, hasten to your Apostolic See in order to receive from you a cure for the wounds of the churches.

^{Epistula 113 ad Leonem I.}

Prosper of Aquitaine [who] says in his Chronicle for the year 420

In a council held at Carthage consisting of 216 bishops, the synodal decrees were advanced to Pope Zozimus; after they were approved, the Pelagian heresy was condemned throughout the world.

Thus, the whole world recognizes a final judgment only from the Roman Pontiff.

St. Robert also cites:

• Innocent I, in epistola ad Anastasium Thessalonicensem
• Gelasius I epistola ad Episcopos Dardaniæ
• Nicolas I epistola ad Michaelem Imperatorem
• Innocent III epistola ad archatensem Episcopum, there the chapter is extent “Majores”, and the one on Baptism.

Prefacing his dogmatic definition of papal infallibility, Pius IX mentions in particular (Pastor Æternus ch. 4):

the Fathers of the Fourth Council of Constantinople, [who,] following in the footsteps of their predecessors, gave forth this solemn profession: The first condition of salvation is to keep the rule of the true faith. And because the sentence of our Lord Jesus Christ can not be passed by, who said: “Thou art Peter, and upon this rock I will build my Church,” [Mt. 16:18] these things which have been said are approved by events, because in the Apostolic See the Catholic religion and her holy and well-known doctrine has always been kept undefiled. Desiring therefore, not to be in the least degree separated from the faith and doctrine of that See, we hope that we may deserve to be in the one communion, which the Apostolic See preaches, in which is the entire and true solidity of the Christian religion.

^{From the Formula of St. Hormisdas, subscribed by the Fathers of the Eighth General Council (Fourth of Constantinople), A.D. 869 (Labbe's Councils, Vol. V. pp. 583, 622).}

He also cites early 5^th century precedent for assembling in council to define/clarify matters of faith:

the Bishops of the whole world, now singly, now assembled in Synod, following the long-established custom of churches,* and the form of the ancient rule,^† sent word to this Apostolic See of those dangers especially which sprang up in matters of faith, that there the losses of faith might be most effectually repaired where the faith can not fail.

^{*From a letter of St. Cyril of Alexandria to Pope St. Celestine I., A.D. 422 (Vol. VI. Part II. p. 36, Paris edition of 1638).
†From a Rescript of St. Innocent I. to the Council of Milevis, A.D. 402 (Labbe, Vol. III. p. 47).}

I found an official Roman Catholic web site with an article entitled ‘Did the early church fathers espouse papal infallibility?’ The first article (link below) opened with this observation:

The infallibility of the Church in general—and the pope in particular—is not a doctrine that suddenly appeared in Church teaching; rather, it is a doctrine that was affirmed in various ways in the early Church...

The article went on to explain:

As Christians began to more clearly understand the teaching authority of the Church and of the primacy of the pope, they developed a clearer understanding of the pope’s infallibility. For example, in the late second century, in his magnum opus Against the Heresies, St. Irenaeus of Lyons, writes of the Church of Rome, over which Peter and his papal successors preside: With that church, because of its superior origin, all the churches must agree, that is, all the faithful in the whole world, and it is in her that the faithful everywhere have maintained the apostolic tradition ((Against Heresies 3:3:2 [A.D. 189]).

In 251, St. Cyprian of Carthage, writes: If someone does not hold fast to this unity of Peter, can he imagine that he still holds the faith? If he [should] desert the chair of Peter upon whom the Church was built, can he still be confident that he is in the Church? (The Unity of the Catholic Church 4; 1st edition [A.D. 251]).

Further, in the fifth century, St. Augustine succinctly captured the ancient attitude when he remarked, “Rome has spoken; the case is concluded” (Sermons 131, 10). Source: https://www.catholic.com/qa/did-the-early-church-fathers-espouse-papal-infallibility

From there I found a linked article giving the views of the popes and other church fathers up to the year A.D. 345:

In a wide variety of ways, the Fathers attest to the fact that the church of Rome was the central and most authoritative church. They attest to the Church’s reliance on Rome for advice, for mediation of disputes, and for guidance on doctrinal issues. They note, as Ignatius of Antioch does, that Rome “holds the presidency” among the other churches, and that, as Irenaeus explains, “because of its superior origin, all the churches must agree” with Rome. They are also clear on the fact that it is communion with Rome and the bishop of Rome that causes one to be in communion with the Catholic Church. This displays a recognition that, as Cyprian of Carthage puts it, Rome is “the principal church, in which sacerdotal unity has its source. Source: https://www.catholic.com/tract/the-authority-of-the-pope-part-i

A follow-on article covered the views of the fathers after A.D. 341:

Jerome: “I follow no leader but Christ and join in communion with none but your blessedness [Pope Damasus I], that is, with the chair of Peter. I know that this is the rock on which the Church has been built. Whoever eats the Lamb outside this house is profane. Anyone who is not in the ark of Noah will perish when the flood prevails” (Letters 15:2 [A.D. 396]).

Augustine: “[On this matter of the Pelagians] two councils have already been sent to the Apostolic See [the bishop of Rome], and from there rescripts too have come. The matter is at an end; would that the error too might be at an end!” (Sermons 131:10 [A.D. 411]).

Peter Chrysologus: “We exhort you in every respect, honorable brother, to heed obediently what has been written by the most blessed pope of the city of Rome, for blessed Peter, who lives and presides in his own see, provides the truth of faith to those who seek it. For we, by reason of our pursuit of peace and faith, cannot try cases on the faith without the consent of the bishop of Rome” (Letters 25:2 [A.D. 449]).

Council of Chalcedon: “After the reading of the foregoing epistle [The Tome of Leo], the most reverend bishops cried out: ‘This is the faith of the fathers! This is the faith of the apostles! So we all believe! Thus the orthodox believe! Anathema to him who does not thus believe! Peter has spoken thus through Leo!’” (ibid., session 2). Source: https://www.catholic.com/tract/the-authority-of-the-pope-part-ii

I understand that the Catholic Church views encyclicals and dogma as inerrant because they view the church as indefectible. In other words, the church is not subject to failure or decay, has no defects and is perfect. If I’ve got that wrong, will someone please correct me?

Conclusion: Whilst all of the quotes affirm the beliefs of the early church fathers with regard to the teaching authority of the church, papal authority and the primacy of the pope, nowhere could I find any quote suggesting they believed that papal infallibility had been conferred upon any pope. Again, if I am mistaken, I hope to be corrected.