Sort of. Tacitus, for example, mentions his execution a hundred years later but his cult was relatively popular, spreading throughout the Jewish diaspora, having a notable presence in Rome itself, and presumably played a part in a massive rebellion in Judaea. Of course, a lot of the Roman sources could also be fabricated references, but Tacitus’s is considered reliable.

There isn’t a document saying “Pilate had this dude executed today” but there are sources saying “That dude Pilate executed back in the day has a cult and they’re being annoying.”

So yes, there almost certainty was a cult leader that was executed and his followers believed he was divine, or at least started saying he was within a couple decades of his death, but we, for example, don’t even have a primary source that his name was Yeshua. We mostly know for certain that people worshipped a guy named Yeshua a century after his probable death.

But history is funny, and cults are weird.

It’s not impossible, for example, a man with a cult named something too “ethnic” for Greeks and Romans to pronounce was stabbed by a Roman by the side of the road. They scatter and start telling everyone about how a soldier of Pilate martyred their God, it becomes a crucifixion over time where a soldier stabs him in the side, and everyone gives up on correcting people that his name wasn’t “Josh” and just rolls with it because at least they’re getting the “Christ” title right.

But we don’t have a source for that version, so you can broadly assume a dude named Jesus was baptized by John, crucified by Pilate, and these were reliable events even if everything else isn’t.