While there are no early Arab texts about the life of the prophet of Islam, we do have copies of non-Muslim (mainly Christian) writings that reference the Arab conquest of the Middle East. These include eye witness testimony to the events that take place in the 2nd and 3rd quarters of the 7th century. While the authors provide limited details on the Arabs that conquered Syria, Palestine, and Egypt, they do provide some interesting insights into how the conquered people of those areas viewed their new rulers. In particular, a few references to a "Muhammad" can be found which dates to the first few decades of the Arab conquests.
Doctrina Jacobi
[Jacob, himself a convert, wrote to encourage Christian faith in Jews of Carthage, forcibly converted in 632, in a tract that was completed before "the thirteenth of July in the seventh indiction," i.e. 634, when Jacob left Carthage. In it his cousin Justus appears telling how he heard of the killing of a member of the imperial guard, or candidatus, in a letter from his brother Abraham in Caesarea, in which the following appears.]
When the Candidatus was killed by the Saracens, I was at Caesarea and I set off by boat to Sykamina. People were saying "the candidatus has been killed," and we Jews were overjoyed. And they were saying that the prophet had appeared, coming with the Saracens, and that he was proclaiming the advent of the anointed one, the Christ who was to come. I, having arrived at Sykamina, stopped by a certain old man well-versed in scriptures, and I said to him: "What can you tell me about the prophet who has appeared with the Saracens?" He replied, groaning deeply: "He is false, for the prophets do not come armed with a sword. Truly they are works of anarchy being committed today and I fear that the first Christ to come, whom the Christians worship, was the one sent by God and we instead are preparing to receive the Antichrist. Indeed, Isaiah said that the Jews would retain a perverted and hardened heart until all the earth should be devastated. But you go, master Abraham, and find out about the prophet who has appeared." So I, Abraham, inquired and heard from those who had met him that there was no truth to be found in the so-called prophet, only the shedding of men's blood. He says also that he has the keys of paradise, which is incredible.
[1] Doctrina Jacobi V.16, 209. [p. 57]
Syriac Gospel Fragment
In January {the people of} Hims took the word for their lives and many villages were ravaged by the killing of {the Arabs of} Muhammad (Muhmd) and many people were slain and {taken} prisoner from Galilee as far as Beth...
[1] Syriac Gospel Fragment
Sophronius
But the present circumstances are forcing me to think differently about our way of life, for why are [so many] wars being fought among us? Why do barbarian raids abound? Why are the troops of the Saracens attacking us? Why has there been so much destruction and plunder? Why are there incessant outpourings of human blood? Why are the birds of the sky devouring human bodies? Why have churches been pulled down? Why is the cross mocked? Why is Christ, who is the dispenser of all good things and the provider of this joyousness of ours, blasphemed by pagan mouths (ethnikois tois stomasi) so that he justly cries out to us: "Because of you my name is blasphemed among the pagans," and this is the worst of all the terrible things that are happening to us. That is why the vengeful and God-hating Saracens, the abomination of desolation clearly foretold to us by the prophets, overrun the places which are not allowed to them, plunder cities, devastate fields, burn down villages, set on fire the holy churches, overturn the sacred monasteries, oppose the Byzantine armies arrayed against them, and in fighting raise up the trophies [of war] and add victory to victory. Moreover, they are raised up more and more against us and increase their blasphemy of Christ and the church, and utter wicked blasphemies against God. Those God-fighters boast of prevailing over all, assiduously and unrestrainably imitating their leader, who is the devil, and emulating his vanity because of which he has been expelled from heaven and been assigned to the gloomy shades.
Sophronius Holy Baptism, 166-167
Thomas the Presbyter
In the year 945, indiction 7, on Friday 7 February (634) at the ninth hour, there was a battle between the Romans and the Arabs of Muhammad (tayyaye d-Mhmt) in Palestine twelve miles east of Gaza. The Romans fled, leaving behind the patrician bryrdn, whom the Arabs killed. Some 4000 poor villagers of Palestine were killed there, Christians, Jews and Samaritans. The Arabs ravaged the whole region.
[1] Thomas the Presbyter, Chronicle, pp. 147-148
Sebeos, Bishop Of The Bagratunis
So [The Jews] departed, taking the road through the desert to Tachkastan to the sons of Ishmael. [The Jews] called [the Arabs] to their aid and familiarized them with the relationship they had through the books of the [Old] Testament. Although [the Arabs] were convinced of their close relationship, they were unable to get a consensus from their multitude, for they were divided from each other by religion. In that period a certain one of them, a man of the sons of Ishmael named Muhammad, became prominent [t'ankangar]. A sermon about the Way of Truth, supposedly at God's command, was revealed to them, and [Muhammad] taught them to recognize the God of Abraham, especially since he was informed and knowledgeable about Mosaic history. Because the command had [g104] come from on High, he ordered them all to assemble together and to unite in faith. Abandonning the reverence of vain things, they turned toward the living God, who had appeared to their father--Abraham. Muhammad legislated that they were not to [123] eat carrion, not to drink wine, not to speak falsehoods, and not to commit adultery. He said: "God promised that country to Abraham and to his son after him, for eternity. And what had been promised was fulfilled during that time when [God] loved Israel. Now, however, you are the sons of Abraham, and God shall fulfill the promise made to Abraham and his son on you. Only love the God of Abraham, and go and take the country which God gave to your father Abraham. No one can successfully resist you in war, since God is with you."
[1] Sebeos Ch 30
John, Bishop of Nikiu
And now many of the Egyptians who had been false Christians denied the holy orthodox faith and lifegiving baptism, and embraced the religion of the Moslem, the enemies of God, and accepted the detestable doctrine of the beast, this is, Mohammed, and they erred together with those idolaters, and took arms in their hands and fought against the Christians. And one of them, named John, the Chalcedonian of the Convent of Sinai, embraced the faith of Islam, and quitting his monk's habit he took up the sword, and persecuted the Christians who were faithful to our Lord Jesus Christ.
[1] John Chronicle, 121.10-11