Polycarp resided in Asia Minor as bishop of Smyrna and sent an epistle to the Philippians c. 120-140 CE. Polycarp was martyred c. 155 CE.
In Adv. Haer. V.33.4, Irenaeus describes Papias as "the hearer of John, and a companion of Polycarp."
Irenaeus mentions Polycarp in Adv. Haer., III.3.4.
But Polycarp also was not only instructed by apostles, and conversed with many who had seen Christ, but was also, by apostles in Asia, appointed bishop of the Church in Smyrna, whom I also saw in my early youth, for he tarried [on earth] a very long time, and, when a very old man, gloriously and most nobly suffering martyrdom, departed this life, having always taught the things which he had learned from the apostles, and which the Church has handed down, and which alone are true. To these things all the Asiatic Churches testify, as do also those men who have succeeded Polycarp down to the present time, a man who was of much greater weight, and a more stedfast witness of truth, than Valentinus, and Marcion, and the rest of the heretics. He it was who, coming to Rome in the time of Anicetus caused many to turn away from the aforesaid heretics to the Church of God, proclaiming that he had received this one and sole truth from the apostles, that, namely, which is handed down by the Church. There are also those who heard from him that John, the disciple of the Lord, going to bathe at Ephesus, and perceiving Cerinthus within, rushed out of the bath-house without bathing, exclaiming, "Let us fly, lest even the bath-house fall down, because Cerinthus, the enemy of the truth, is within." And Polycarp himself replied to Marcion, who met him on one occasion, and said, "Dost thou know me? "I do know thee, the first-born of Satan." Such was the horror which the apostles and their disciples had against holding even verbal communication with any corrupters of the truth; as Paul also says, "A man that is an heretic, after the first and second admonition, reject; knowing that he that is such is subverted, and sinneth, being condemned of himself." There is also a very powerful Epistle of Polycarp written to the Philippians, from which those who choose to do so, and are anxious about their salvation, can learn the character of his faith, and the preaching of the truth. Then, again, the Church in Ephesus, founded by Paul, and having John remaining among them permanently until the times of Trajan, is a true witness of the tradition of the apostles.
Irenaeus also mentions Polycarp in his letter to Florinus.
These opinions, Florinus, that I may speak in mild terms, are not of sound doctrine; these opinions are not consonant to the Church, and involve their votaries in the utmost impiety; these opinions, even the heretics beyond the Church's pale have never ventured to broach; these opinions, those presbyters who preceded us, and who were conversant with the apostles, did not hand down to thee. For, while I was yet a boy, I saw thee in Lower Asia with Polycarp, distinguishing thyself in the royal court, and endeavouring to gain his approbation. For I have a more vivid recollection of what occurred at that time than of recent events (inasmuch as the experiences of childhood, keeping pace with the growth of the soul, become incorporated with it); so that I can even describe the place where the blessed Polycarp used to sit and discourse-his going out, too, and his coming in-his general mode of life and personal appearance, together with the discourses which he delivered to the people; also how he would speak of his familiar intercourse with John, and with the rest of those who had seen the Lord; and how he would call their words to remembrance. Whatsoever things he had heard from them respecting the Lord, both with regard to His miracles and His teaching, Polycarp having thus received [information] from the eye-witnesses of the Word of life, would recount them all in harmony with the Scriptures. These things, through, God's mercy which was upon me, I then listened to attentively, and treasured them up not on paper, but in my heart; and I am continually, by God's grace, revolving these things accurately in my mind. And I can bear witness before God, that if that blessed and apostolical presbyter had heard any such thing, he would have cried out, and stopped his ears, exclaiming as he was wont to do: "O good God, for what times hast Thou reserved me, that I should endure these things? "And he would have fled from the very spot where, sitting or standing, he had heard such words. This fact, too, can be made clear, from his Epistles which he despatched, whether to the neighbouring Churches to confirm them, or to certain of the brethren, admonishing and exhorting them.
Irenaeus also mentions Polycarp in his letter to Pope Victor.
For the controversy is not merely as regards the day, but also as regards the form itself of the fast. For some consider themselves hound to fast one day, others two days, others still more, while others [do so during] forty: the diurnal and the nocturnal hours they measure out together as their [fasting] day. And this variety among the observers [of the fasts] had not its origin in our time, but long before in that of our predecessors, some of whom probably, being not very accurate in their observance of it, handed down to posterity the custom as it had, through simplicity or private fancy, been [introduced among them]. And yet nevertheless all these lived in peace one with another, and we also keep peace together. Thus, in fact, the difference [in observing] the fast establishes the harmony of [our common] faith. And the presbyters preceding Sorer in the government of the Church which thou dost now rule-I mean, Anicetus and Pius, Hyginus and Telesphorus, and Sixtus-did neither themselves observe it [after that fashion], nor permit those with them to do so. Notwithstanding this, those who did not keep [the feast in this way] were peacefully disposed towards those who came to them from other dioceses in which it was [so] observed (although such observance was [felt] in more decided contrariety [as presented] to those who did not fall in with it; and none were ever cast out [of the Church] for this matter. On the contrary, those presbyters who preceded thee, and who did not observe [this custom], sent the Eucharist to those of other dioceses who did observe it. And when the blessed Polycarp was sojourning in Rome in the time of Anicetus, although a slight controversy had arisen among them as to certain other points, they were at once well inclined towards each other [with regard to the matter in hand], not willing that any quarrel should arise between them upon this head. For neither could Anicetus persuade Polycarp to forego the observance [in his own way], inasmuch as these things had been always [so] observed by John the disciple of our Lord, and by other apostles with whom he had been conversant; nor, on the other hand, could Polycarp succeed in persuading Anicetus to keep [the observance in his way], for he maintained that he was bound to adhere to the usage of the presbyters who preceded him. And in this state of affairs they held fellowship with each other; and Anicetus conceded to Polycarp in the Church the celebration of the Eucharist, by way of showing him respect; so that they parted in peace one from the other, maintaining peace with the whole Church, both those who did observe [this custom] and those who did not.
Jerome provides the following summary in Illustrious Men 17.
Polycarp disciple of the apostle John and by him ordained bishop of Smyrna was chief of all Asia, where he saw and had as teachers some of the apostles and of those who had seen the Lord. He, on account of certain questions concerning the day of the Passover, went to Rome in the time of the emperor Antoninus Pius while Anicetus ruled the church in that city. There he led back to the faith many of the believers who had been deceived through the persuasion of Marcion and Valentinus, and when. Marcion met him by chance and said "Do you know us" he replied, "I know the firstborn of the devil." Afterwards during the reign of Marcus Antoninus and Lucius Aurelius Commodus in the fourth persecution after Nero, in the presence of the proconsul holding court at Smyrna and all the people crying out against him in the Amphitheater, he was burned. He wrote a very valuable Epistle to the Philippians which is read to the present day in the meetings in Asia.
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