Absolutely contemporaneous, as in written before he was crucified? No. For that matter, we don't have any such evidence for Julius Caesar either, except for the buildings and coins he commissioned; or for Pontius Pilate except for one inscription-stone. (EDIT: Okay, Caesar's mentioned in Cicero's letters; he had the advantage of interacting with influential writers. The comparison still stands for Pilate, and for Caiaphas too apart from his ossuary.) The lack of absolutely-contemporaneous evidence is completely normal for figures from ancient history. What would be surprising is if we did have such evidence for someone who hardly ever left Galilee and Judea, commissioned no buildings, minted no coins, and attracted no public following among the upper classes.

We do, however, have evidence from shortly after Jesus' crucifixion. (Again, that's "shortly after" as used in ancient history; we don't have evidence from the first five years or so, nor would we expect to.) The dates of the Gospels are disputed, but they were definitely composed within fifty years or so afterwards. Also, here're some threads about early extra-Biblical evidence. All this's totally normal; our surviving biographies of other ancient figures have similar or even worse timing.

If you've got any other questions, don't hesitate to ask!