Is there any early non-Christian evidence that Jesus existed?

Yes, there is. Some of the evidence is mentioned below.

The Jewish historian Josephus mentions Jesus at least once in his Antiquities of the Jews (book 20, chapter 9; c. 93 AD):

…Festus was now dead, and Albinus was but upon the road; so he [Ananus] assembled the sanhedrim of judges, and brought before them the brother of Jesus, who was called Christ, whose name was James, and some others; and when he had formed an accusation against them as breakers of the law, he delivered them to be stoned…¹

There is also a reference to Jesus in Josephus’ Antiquities of the Jews book 18, chapter 3:

Now there was about this time Jesus, a wise man, if it be lawful to call him a man; for he was a doer of wonderful works, a teacher of such men as receive the truth with pleasure. He drew over to him both many of the Jews and many of the Gentiles. He was [the] Christ. And when Pilate, at the suggestion of the principal men amongst us, had condemned him to the cross, those that loved him at the first did not forsake him; for he appeared to them alive again the third day; as the divine prophets had foretold these and ten thousand other wonderful things concerning him. And the tribe of Christians, so named from him, are not extinct at this day.^{2, 3}

Mara Bar-Serapion, writing to his son possibly c. 73 AD, talks of a ‘wise king’ of the Jews:

For what benefit did the Athenians obtain by putting Socrates to death, seeing that they received as retribution for it famine and pestilence? Or the people of Samos by the burning of Pythagoras, seeing that in one hour the. whole of their country was covered with sand? Or the Jews by the murder of their Wise King, seeing that from that very time their kingdom was driven away from them? For with justice did God grant a recompense to the wisdom of all three of them. For the Athenians died by famine; and the people of Samos were covered by the sea without remedy; and the Jews, brought to desolation and expelled from their kingdom, are driven away into Every land. Nay, Socrates did “not” die, because of Plato; nor yet Pythagoras, because of the statue of Hera; nor yet the Wise King, because of the new laws which he enacted.⁴

Reference to a Jewish “wise king” (cf. Mt. 2:2; 27:11, 29, 37; Mk. 15:2, 9, 12, 18, 26; Lk. 23:3, 37, 38, 51; Jn. 18:33, 36, 39; 19:3, 12, 14, 19, 21) — one who, shortly before the Jews were exiled from Israel, was killed by the his countrymen for teaching ‘new laws’ — fits no other individual in history but Jesus.

The Roman historian Tacitus wrote in his Annals (109 AD), book 15:

…Nero fastened the guilt and inflicted the most exquisite tortures on a class hated for their abominations, called Christians by the populace. Christus, from whom the name had its origin, suffered the extreme penalty during the reign of Tiberius at the hands of one of our procurators, Pontius Pilatus, and a most mischievous superstition, thus checked for the moment, again broke out not only in Judaea, the first source of the evil, but even in Rome, where all things hideous and shameful from every part of the world find their centre and become popular.⁵

Suetonius, in Life of Claudius (121 AD), chapter 25, wrote:

He banished from Rome all the Jews, who were continually making disturbances at the instigation of one Chrestus.⁶

Lucian (b. 125), in The Death of Peregrine, wrote:

The Christians, you know, worship a man to this day,–the distinguished personage who introduced their novel rites, and was crucified on that account. […] it was impressed on them by their original lawgiver that they are all brothers, from the moment that they are converted, and deny the gods of Greece, and worship the crucified sage, and live after his laws.⁷

There are some texts that mention Jesus that have been lost in history. We know of these texts because we have the writings of later people which mention them. For example, Justin Martyr, in his First Apology (c. 150 AD), refers to Roman documents that record aspects of Jesus’ life:

And the expression, “They pierced my hands and my feet,” was used in reference to the nails of the cross which were fixed in His hands and feet. And after He was crucified they cast lots upon His vesture, and they that crucified Him parted it among them. And that these things did happen, you can ascertain from the Acts of Pontius Pilate.⁸

In a chapter earlier (ch.34) he says:

Now there is a village in the land of the Jews, thirty-five stadia from Jerusalem, in which Jesus Christ was born, as you can ascertain also from the registers of the taxing made under Cyrenius…⁸

Likewise, Tertullian, in Against Maricon (c. 208 AD), refers to official Roman documents, which are now lost, that mentioned Jesus:

…the Roman registry still has in keeping, a most faithful witness to our Lord’s nativity…⁹

Origen quotes from a 2^nd century philosopher called Celsus , who was an opponent of Christianity, in a rebuttal of him (Against Celsus; 248 AD). Celsus denies that Jesus is the Christ, the son of God; but he doesn’t deny that Jesus existed. Three examples of quotes from Celsus are:

[Jesus] invented his birth from a virgin … [Jesus was actually] born in a certain Jewish village, of a poor woman of the country, who gained her subsistence by spinning, and who was turned out of doors by her husband, a carpenter by trade, because she was convicted of adultery; that after being driven away by her husband, and wandering about for a time, she disgracefully gave birth to Jesus, an illegitimate child, who having hired himself out as a servant in Egypt on account of his poverty, and having there acquired some miraculous powers, on which the Egyptians greatly pride themselves, returned to his own country, highly elated on account of them, and by means of these proclaimed himself a God.
(Against Celsus, book 1, chapter 28)¹⁰

…we visited with punishment the man who deluded you…
(book 2, chapter 4)¹¹

By what train of argument were you led to regard him as the Son of God?
(chapter 47)¹²

The fact that Jesus was a historical figure was not under dispute: Jesus existed. And, as Christians, we also believe that he still exists.

Consequently, he is able to save to the uttermost those who draw near to God through him, since he always lives to make intercession for them. (Heb. 7:25)

————

References

1. Josephus, Antiquities of the Jews (c. 93 AD), book 20, chapter 9; online here (accessed 18/11/09)
2. Josephus, Antiquities of the Jews (c. 93 AD), book 18, chapter 3; online here (accessed 18/11/09)
3. There is a possible problem with this passage, though: some of the words might not have be written by Josephus, they might have been added later. It has been suggested that the following reconstruction is possibly closer to the original:
   

   Now there was about this time Jesus, a wise man, for he was a doer of wonders. He drew many after him. When Pilate, at the suggestion of the principal men among us, had condemned him to the cross, those that loved him at the first did not forsake him, and the tribe of Christians, so named from him, are not extinct at this day

4. Mara Bar-Serapion, A Letter of Mara, Son of Serapion (c. 73 AD); online here (accessed 18/11/09)
5. Tacitus, The Annals (109 AD), book 15; online here (accessed 18/11/09)
6. Suetonius, Life of Claudius (121 AD), chapter 25; online here (accessed 18/11/09)
7. Lucian (b. 125), The Death of Peregrine; online here (accessed 18/11/09)
8. Justin Martyr, First Apology (c. 150 AD), chapters 34&35; online here (accessed 18/11/09)
9. Tertullia, Against Marcion (c. 208 AD), book 4, chapter 7; online here (accessed 18/11/09)
10. Origen, Against Celsus (248 AD), book 1, chapter 28; online here (accessed 18/11/09)
11. Origen, Against Celsus (248 AD), book 2, chapters 4; online here (accessed 18/11/09)
12. Origen, Against Celsus (248 AD), book 2, chapters 47; online here (accessed 18/11/09)

Further reading

F. F. Bruce, The New Testament Documents: Are they reliable? fifth edn. (Leicester: IVP, 1960), p.100-120