Is there any evidence outside of Islamic sources that show the prophet Mohammed existed?

As you may know already, most of the hadiths, the source of so much of what we 'know' about early Islam, were from the eighth century or later. However, the Qur'an was compiled in the early seventh century, with at least one version being completed in the early 630s (see B Sadeghi, 'The Codex of a Companion of the Prophet and the Qurān of the Prophet', Arabica, Volume 57/4, 2010), so presumably someone was creating a religious movement in Arabia at this point, which is surely something in favour of the Prophet's existence. Even though the Qur'an rarely mentioned Muhammad, he was still nonetheless named, so it can be used as evidence for Muhammad being a political/religious leader at this point. This can perhaps be corroborated by the Constitution of Medina, a piece of information contained in the late eighth-century Sirat of Ibn Ishaq. This technically only survives in the even later works of Ibn Hisham and al-Tabari, but the Arabic used in this passage is archaic, so it is often considered to be a direct survival from the 620s (though I know a few people who do not find this convincing). Again, it preserves no biographical details about Muhammad, but it is evidence that there was a growing community of Believers in Medina presumably under his leadership. As such, I don't think we should dismiss Islamic sources for Muhammad's existence just because they implicitly accept his existence.

In any case, we do have early non-Islamic sources that point to the same. From Roman North Africa, we have what is essentially a Roman propaganda leaflet directed at Jews in Carthage that tried to convince them that they should not listen to all the bad news coming in from the east. The narrative is dated to circa 634 given the things it talked about, but realistically it could have been written at any point in the 630s, as our chronology of this period is still unclear. The relevant passage is this:

Justus answered and said, “Indeed you speak the truth, and this is the great salvation: to believe in Christ. For I confess to you, master Jacob, the complete truth. My brother Abraham wrote to me that a false prophet has appeared. Abraham writes, ‘When [Sergius] the candidatus was killed by the Saracens, I was in Caesarea, and I went by ship to Sykamina. And they were saying, “The candidatus has been killed,” and we Jews were overjoyed. And they were saying, “A prophet has appeared, coming with the Saracens and he is preaching the arrival of the anointed one who is to come, the Messiah.”

And when I arrived in Sykamina, I visited an old man who was learned in the scriptures, and I said to him, “What can you tell me about the prophet who has appeared with the Saracens?” And he said to me, groaning loudly, “He is false, for prophets do not come with a sword and a war-chariot. Truly the things set in motion today are deeds of anarchy, and I fear that somehow the first Christ that came, whom the Christians worship, was the one sent by God, and instead of him we will receive the Antichrist. Truly, Isaiah said that we Jews will have a deceived and hardened heart until the entire earth is destroyed. But go, master Abraham, and find out about this prophet who has appeared.” And when I, Abraham, investigated thoroughly, I heard from those who had met him that one will find no truth in the so-called prophet, only the shedding of human blood. In fact, he says that he has the keys of paradise, which is impossible.’ These things my brother Abraham has written from the East.”

Obviously, the author of the pamphlet was not particularly interested in being super-accurate, but his words are nonetheless still an indication of what Romans in Carthage thought about this new crisis in the east: that a prophet was involved, that he claimed to be a herald of the apocalypse, and that the Romans were worried about the Jews being perhaps a fifth-column for these Arabs. These things all fit very well with we know from the Qur'an and other later sources, so I think these claims are quite reasonable. Though he was not named in this pamphlet, Muhammad's existence fits quite well with the evidence provided by this extraordinary document.

There are two other Syriac sources from the late 630s as well, though they both tell us very little about Muhammad. From a faded note in a Bible circa 637:

. . . Muhammad . . . priest, Mār Elijah . . . and they came . . . and . . . and from . . . strong . . . month . . . and the Romans {fled} . . . And in January {the people} of Emesa received assurances for their lives. Many villages were destroyed through the killing by {the Arabs of} Muhammad and many people were killed. And captives {were taken} from the Galilee to Bēt . . . Those Arabs camped by {Damascus}. We saw . . . everywhere . . . and the {olive oil} that they {had brought} and . . . them. On the twenty-sixth of May, {the sacellarius} went . . . from Emesa. The Romans pursued them . . . On the tenth {of August} . . . the Romans fled from Damascus . . . many, about ten thousand. The following year, the Romans came. On the twentieth of August in the year nine hundred and forty-seven [636 c.e.] there assembled in Gabitha . . . the Romans and many people were killed, from the Romans about fifty thousand . . . In the year nine hundred and forty-{eight} . . .

... and from the Chronicle to 640:

In the year 945 [634], the seventh indiction, on Friday, February the fourth, at the ninth hour, there was a battle between the Romans and the Arabs of Muhammad in Palestine, twelve miles east of Gaza. The Romans fled. They abandoned the patrician Bryrdn, and the Arabs killed him. About four thousand poor villagers from Palestine—Christians, Jews, and Samaritans— were killed, and the Arabs destroyed the whole region.

These sources imply that Muhammad personally led attacks into Palestine, but they might also just mean that Muhammad played an important role in Arab society even after his death. Either way, they attest to his existence, though it is certainly true that they are evidence that the traditional narrative of Muhammad's career is flawed. Stephen Shoemaker for example explored this in his The Death of a Prophet: The End of Muhammad's Life and the Beginnings of Islam (2011), using early sources to argue that Muhammad was alive after 632 (the traditional date for his death) and that he played a leading role in the raids against the Roman empire. These suggestions are not at all certain and scholars are still debating their veracity, but these early attestations to Muhammad are surely notable - especially as these authors were evidently not inclined to be kind to the Prophet!

By the 660s, we have our first detailed narrative of Muhammad's life, this time from an Armenian source, pseudo-Sebeos:

At that time a certain man from among those same sons of Ishmael whose name was Mahmet, a merchant, as if by God’s command appeared to them as a preacher [and] the path of truth. He taught them to recognize the God of Abraham, especially because he was learned and informed in the history of Moses. Now because the command was from on high, at a single order they all came together in unity of religion. Abandoning their vain cults, they turned to the living God who had appeared to their father Abraham. So Mahmet legislated for them: not to eat carrion, not to drink wine, not to speak falsely, and not to engage in fornication.

He said: ‘With an oath God promised this land to Abraham and his seed after him for ever. And he brought about as he promised during that time while he loved Israel. But now you are the sons of Abraham, and God is accomplishing his promise to Abraham and his seed for you. Love sincerely only the God of Abraham, and go and seize your land which God gave to your father Abraham. No one will be able to resist you in battle, because God is with you.'

This is essentially the gist of the narrative preserved in later Muslim sources, so it does suggest that within ~30 years of Muhammad's death the details of his life were being noted down and spread across the Middle East, even reaching the library of an author deeply hostile to the Arabs. Of course, even within a short period of time the story was no doubt mythologised to an extent, but pseudo-Sebeos' tale also preserved more archaic elements of the story, as he tied early Islam closely to Judaism, whereas later sources tended to minimise this aspect, so I think it is likely that part of this narrative is essentially accurate. The same is true with Muhammad's life more generally. Western Arabia simply wasn't that big a deal, so why would anyone want to write about it? Our sources for the world beyond it is pretty bad too - there are basically only two Greek historical texts from this period (both of which stopped c.630) and none from Persia. For most events we have to rely on fragmentary Syriac and Coptic sources, as well letters and sermons from ecclesiastical writers. The fact that we have a plethora of contemporaries writing about Muhammad's enormous influence is therefore testament to his importance. He may have been a very different man to the prophet described in later narratives, but he certainly existed.

You might also be interested to read my answer on early Christian reactions to Islam; though I did not discuss Muhammad, the existence of a new leader on the fringes of two world empires makes sense given the evidence available, if only because of the sudden rise of the caliphate. How else are we to explain its rise, if not the existence of a politically astute leader who had preached a very late antique message?