B.M.
Hi N.,
My daughter was exactly like that at that age (she is now 9). She started around 3 or 4!!!!! In her case it's TOTALLY because she is a perfectionist and I'd LOVE to tell you I don't have any idea where she gets that from :-)
What worked for us (and may not work for everyone) is I started giving her "practice goals". At first they were just ridiculous - so for the sign language I'd say "this is really hard and I would say you should have 15 practice tries before you give up". Sometimes after I would see if it was difficult for her on the first try or if it was something I knew she might struggle with I would say "I don't expect you to get this anywhere close for 10 tries" or whatever. It worked wonders almost immediately because it gave her #1 the OK to fail and #2 the motivation to try again to get it right. For her, it also meant she wouldn't "disappoint me" and it totally took the pressure off.
After a while (and by "a while" I mean a year or so) of me giving her the "practice goals" she just sort of got the hang of it and naturally does it. if I asked her pick up her toys or whatever task, she would go do it and then she would come and get me to have a "preliminary look".
Also around 5 or 6 we transitioned to where I would ask her how many times she thought she would need to "practice" after we set task or whatever. Sometimes she still needs help managing it, but for the most part it has been a WONDERFUL fix.
Let me also say that this gave her a PERFECT set up when she started writing at school to learn how to write rough drafts, happens about 2nd grade or so and most kids struggle with at first because they don't want to revise their work when they are young. But when she wanted to practice writing I would give her 3 sheets of paper and say "I don't expect any of these to be what you finally turn in - when you have practiced on these 3, then come and get the "good" paper for your final draft". Her teacher actually said to me she is one of the best at 'brainstorming' and 'rough draft revision' because she tells the other kids "that's where you get to make all your mistakes so you get it right in the end".
PS - I also did this when she started to do 'activities' because she wanted to quit right away if she wasn't as good as the other kids. So, same thing - you need to do this session of gymnastics (10 lessons or whatever the term is for the activity) as "practice".
GOOD LUCK!