My DD1 was like this but even more extreme as a child. Birthday parties were always a trial and the photos of her own birthday teas with two or three friends are all of her turned away from the cake burying her face in me. We aren't allowed to sing Happy Birthday to her ever.
She has always hated being the centre of attention and would do anything to avoid being talked to by a teacher or unfamiliar adult. School was quite an ordeal, as were Brownies, dance classes, swimming lessons, etc. I wish these days that we hadn't tried so hard to keep her in those groups.
This behaviour all made more sense when we found out that she has Asperger's Syndrome (not suggest that your DD has).
We always tried to accept her shyness and made a point of not pushing her because I remember what that felt like as a shy child. After the diagnosis we backed off even more and found ways to make it possible for her to remain in control of her own interactions. That seems to have been the right thing to do.
She's now 16 and her confidence has recently increased. She has found friends who accept her for what she is and she is comfortable in her own niche. She socialises in small groups and has a date for her prom later this week. She'll never be the life and soul of the party but she has found a way to fit in socially which suits her.
Today she went to our local town with four friends, had some lunch, bought a couple of things for her prom and hung around just enjoying each other's company for a while. If you'd told me three years ago that she would be doing that I probably would have decked you for taking the p* out of her. I truly never thought I'd see it happen.
Give your DD time and support. Don't push her to do things which don't feel comfortable and help her to see that she doesn't have to be the centre of attention to make a positive contribution. If she wants to sit and watch the others play at birthday parties let her do just that. Hopefully, one day, you'll watch her blossom into a beautiful 16 year old who's looking forward to her prom too .