What are the primary sources for Claudia Procula (aka Mrs. Pilate)?

Factual Questions
1

Pilate’s wife is mentioned briefly in the Gospel of Matthew, though not by name (and she doesn’t make an actual appearance but sends her husband a letter). However, she is given the name Claudia Procula in most accounts, is a saint (along with her mother, Julia) in Eastern Orthodox and (along with her husband) in the Coptic church. In some accounts she is referred to as an illegitimate descendant of Augustus and a relative (due to her gens nomen, I suppose) of the Claudian line that included Tiberius, Claudius, Caligula and Nero.

However, I haven’t found any primary sources identifying Pilate’s wife as Claudia Procula or mentioning her. Is she purely legendary/apocryphal/traditional or is she mentioned as his wife by a 1st century historian? I haven’t found her in Tacitus or Josephus (though I’m not claiming she isn’t there.) Any info appreciated.

2

I was wondering this also, since someone just mentioned to me that she was supposedly the illegitimate daughter of Tiberius. I can find no source other than fanciful historical fiction to back this up.

3

For what it’s worth, she’s also recognized as a saint by the Roman Catholic church (though Pilate himself is not).

I don’t know of any cite for her name, though.