Had similar issues with my son when he was younger – and he seems to have grown out of it.
Advice…
Praise trying – not the results. ie – “You worked really hard on that picture”, instead of “wow, that is a great picture”.
Show and take your own mistakes well. “oh look, I made a mistake on my knitting. I’ll just have to undo it to that point, and try again…” or “I made a mistake here – but it isn’t noticable when you look at the whole sweater…”
When they are saying they can’t do something that you know they can… don’t tell them that they can… to the perfectionist child that ups the anxiety… makes them feel like you expect them to be able to do it… perfectly! so with them saying “I can’t play this song” (even though they have before…) – don’t say “Yes you can, you played it before…” or anything like that. Instead say “Yes, that song can be a little tricky, but all I am asking is that you try”
For learning something, don’t give them a time frame to learn it… again, they are usually so scared of failure that they won’t try.
(My son spent a YEAR in suzuki violin lessons, refusing to play the violin with fingers, or try anything new… it was a long, hard time… and I really had to learn how to deal with the perfectionist child… but I can’t think of anything that he has done this with for a long time. So although it was a long, frustrating, difficult year that often had me in tears – it was worth it!)