I would say you are trying to push too much; that’s what she’s telling you when she refuses to let you try to teach her how to write letters. Your little girl is obviously bright and a good learner, but rare indeed does one find a 3-year-old with the muscular skill and control to handle printing or writing. She could well benefit from a good preschool, but her teachers would certainly not try to teach her to write at her age.
The best way for her to improve her eye-hand coordination—a necessary development before a child can write—is to have plenty of opportunities to play with small toys such as blocks, dolls, housekeeping toys and puzzles. Let her do a lot of art (not coloring books, where she has to try to stay inside the lines) — free-hand drawing and coloring, manipulating clay and finger painting... Along with those opportunities, she needs to enjoy books and the printed word. If you read to her daily and let her point to pictures in the book, before long she will be pointing to letters and saying, “What’s that one?” After that, progress will come fast and furious.
Our parenting advice is given as suggestions only. We recommend you also consult your healthcare provider, and urge you to contact them immediately if your question is urgent or about a medical condition.