How to teach my 4 year old daughter not to be a perfectionist?

[breathe]

My 4 year old is very bright and competitive by nature (I'm sure the competitiveness comes from my genes). I have just started her in piano lessons and I have been helping her at home with her 10 min. daily practicing (I myself am a flute teacher). She is grasping the musical concepts very quickly, but as we work together on her first songs she gets very upset if she makes a single mistake and just gives up. I am very patient with her and have tried to explain that nobody gets it perfect right away and practicing will help her get better (I even told her of my own experiences on the flute), but it hasn't helped. I just want her to enjoy it and feel proud of herself just for trying and to learn how to practice. Any suggestions on how to guide her?
Campinggurl70- Maybe I should clarify that my daughter loves music, loves her piano lessons, but she just gets frustrated at times and I'm working to teach her how to best handle her frustration. I am not pushing her. I am trying to give her some neat experiences and it is my job to help guide her.