The document has been somewhat changed over time, but the base of the story was considered a reliable piece of history, as the people who were present attested to Polycarp's witness of the Christian faith.
Many people who have been martyred for the sake of Christ throughout the ages have had stories told by those who witnessed the events that there was a miraculous presence of God there with signs that accompanied the event.
The piece below is just an excerpt from The Martyrdom of Polycarp, and starts with him entering the stadium to stand his trial of whether he would renounce Christ alone as God or be condemned for his faith.
Now, as Polycarp was entering into the stadium, there came to him a voice from heaven, saying, "Be strong, and show thyself a man, O Polycarp!" No one saw who it was that spoke to him; but those of our brethren who were present heard the voice. And as he was brought forward, the tumult became great when they heard that Polycarp was taken. And when he came near, the proconsul asked him whether he was Polycarp. On his confessing that he was, [the proconsul] sought to persuade him to deny [Christ], saying, "Have respect to thy old age," and other similar things, according to their custom, [such as], "Swear by the fortune of Cæsar; repent, and say, Away with the Atheists." But Polycarp, gazing with a stern countenance on all the multitude of the wicked heathen then in the stadium, and waving his hand towards them, while with groans he looked up to heaven, said, "Away with the Atheists." [Referring the words to the pagans, and not to the Christians, as was desired.] Then, the proconsul urging him, and saying, "Swear, and I will set thee at liberty, reproach Christ;" Polycarp declared, "Eighty and six years have I served Him, and He never did me any injury: how then can I blaspheme my King and my Saviour?"
And when the proconsul yet again pressed him, and said, "Swear by the fortune of Cæsar," he answered, "Since thou art vainly urgent that, as thou sayest, I should swear by the fortune of Cæsar, and pretendest not to know who and what I am, hear me declare with boldness, I am a Christian. And if you wish to learn what the doctrines of Christianity are, appoint me a day, and thou shalt hear them." The proconsul replied, "Persuade the people." But Polycarp said, "To thee I have thought it right to offer an account [of my faith]; for we are taught to give all due honour (which entails no injury upon ourselves) to the powers and authorities which are ordained of God. But as for these, I do not deem them worthy of receiving any account from me."
The proconsul then said to him, "I have wild beasts at hand; to these will I cast thee, except thou repent." But he answered, "Call them then, for we are not accustomed to repent of what is good in order to adopt that which is evil; and it is well for me to be changed from what is evil to what is righteous." But again the proconsul said to him, "I will cause thee to be consumed by fire, seeing thou despisest the wild beasts, if thou wilt not repent." But Polycarp said, "Thou threatenest me with fire which burneth for an hour, and after a little is extinguished, but art ignorant of the fire of the coming judgment and of eternal punishment, reserved for the ungodly. But why tarriest thou? Bring forth what thou wilt." (emphasis mine)
While he spoke these and many other like things, he was filled with confidence and joy, and his countenance was full of grace, so that not merely did it not fall as if troubled by the things said to him, but, on the contrary, the proconsul was astonished, and sent his herald to proclaim in the midst of the stadium thrice, "Polycarp has confessed that he is a Christian." This proclamation having been made by the herald, the whole multitude both of the heathen and Jews, who dwelt at Smyrna, cried out with uncontrollable fury, and in a loud voice, "This is the teacher of Asia, the father of the Christians, and the overthrower of our gods, he who has been teaching many not to sacrifice, or to worship the gods.” Speaking thus, they cried out, and besought Philip the Asiarch to let loose a lion upon Polycarp. But Philip answered that it was not lawful for him to do so, seeing the show of wild beasts were already finished. Then it seemed good to them to cry out with one consent, that Polycarp should be burnt alive. For thus it behooved the vision which was revealed to him in regard to his pillow to be fulfilled, when, seeing it on fire as he was praying, he turned about and said prophetically to the faithful that were with him, "I must be burnt alive."
This, then, was carried into effect with greater speed than it was spoken, the multitudes immediately gathering together wood and fagots out of the shops and baths; the Jews especially, according to custom, eagerly assisting them in it. And when the funeral pile was ready, Polycarp, laying aside all his garments, and loosing his girdle, sought also to take off his sandals, a thing he was not accustomed to do, inasmuch as every one of the faithful was always eager who should first touch his skin. For, on account of his holy life, he was, even before his martyrdom, adorned with every kind of good. Immediately then they surrounded him with those substances which had been prepared for the funeral pile. But when they were about also to fix him with nails, he said, "Leave me as I am; for He that giveth me strength to endure the fire, will also enable me, without your securing me by nails, to remain without moving in the pile."
They did not nail him then, but simply bound him. And he, placing his hands behind him, and being bound like a distinguished ram [taken] out of a great flock for sacrifice, and prepared to be an acceptable burnt-offering unto God, looked up to heaven, and said, "O Lord God Almighty, the Father of thy beloved and blessed Son Jesus Christ, by whom we have received the knowledge of Thee, the God of angels and powers, and of every creature, and of the whole race of the righteous who live before thee, I give Thee thanks that Thou hast counted me, worthy of this day and this hour, that I should have a part in the number of Thy martyrs, in the cup of thy Christ, to the resurrection of eternal life, both of soul and body, through the incorruption [imparted] by the Holy Ghost. Among whom may I be accepted this day before Thee as a fat and acceptable sacrifice, according as Thou, the ever-truthful God, hast foreordained, hast revealed beforehand to me, and now hast fulfilled. Wherefore also I praise Thee for all things, I bless Thee, I glorify Thee, along with the everlasting and heavenly Jesus Christ, Thy beloved Son, with whom, to Thee, and the Holy Ghost, be glory both now and to all coming ages. Amen."
When he had pronounced this amen, and so finished his prayer, those who were appointed for the purpose kindled the fire. And as the flame blazed forth in great fury, we, to whom it was given to witness it, beheld a great miracle, and have been preserved that we might report to others what then took place. For the fire, shaping itself into the form of an arch, like the sail of a ship when filled with the wind, encompassed as by a circle the body of the martyr. And he appeared within not like flesh which is burnt, but as bread that is baked, or as gold and silver glowing in a furnace. Moreover, we perceived such a sweet odor [coming from the pile], as if frankincense or some such precious spices had been smoking there.