DISCLAIMER: I am not a medical professional. I also do not believe that every problem a person may have is a sign of mental illness that needs to be labeled and treated. It can be difficult to untangle symptoms of mental illness from the normal flaws of human nature, and hopefully this student merely has excessively high standards and needs to learn to prioritize better.
Having said that, I’d encourage you to consider the possibility that this student is struggling with obsessive-compulsive disorder (OCD) or a similar condition. I’ve related to a few people (one quite closely) with OCD and similar issues, and what you’ve said about this student sounds very familiar. There are highly effective treatments for OCD, both with and without medication. You can read more about OCD here.
Carolyn outlines a good approach to this kind of problem in her post. One thing I’d add is that it’s important to stay calm about the situation when relating to children who are behaving in these ways. When we get all worked up about this behavior, children may interpret this as a confirmation that they were right to be anxious about this thing, since it’s making the adults anxious too.
When you’re confident that the student has done or can do an assignment well without redoing or checking over anything, you might try suggesting that she hand it in that way and see what grade she gets. This builds the student’s confidence that she can get good results without doing the extra work that she feels is necessary.
Helping children with these issues takes lots of patience and wisdom. May God’s grace be with you!