Yes, there is a higher occurence of perfectionistic tendencies amongst gifted children. However, causality is not well established. It may be an intrinsic tendency, or it may arise out of the higher expectations parents and teachers have of their gifted children - my personal belief is that it probably isn't one or the other, but some combination of both.
There are already some great answers here, so I just want to add that with my own (advanced and perfectionistic child - who is also an only child) as well as with the gifted adolescents I had in my classrooms, I constantly discuss how important mistakes are. Without mistakes we wouldn't learn. If we can already answer all the questions or already do everything exactly right, then we already knew it and aren't learning anything at all. Then I make sure to keep value in learning.
I personally use examples from science because many times scientists learn far more from their mistaken hypotheses than their correct hypotheses (and because I was a science teacher for nearly a decade). However, there are examples of the idea in many walks of life. For example, how many strike outs did Babe Ruth get for every home run? How many publishers turned down your child's favorite books for publishing before the right publishing house was found? Start incorporating as many of these kinds of expamles into her awareness as you can through story and discussion - it will really help in the long run.
Fun Resources that include this concept:
The song (and scene with said song) "up from the ashes" from "Chitty Chitty Bang Bang"
"Meet the Robinsons" (the movie) The idea of "keep moving forward" is illucidated in this movie along with "celebration of failure". Everyone cheers when Louis makes a mistake.
I'm sure there others (I will definitely look for them) and thanks for the question.