I am a swimming teacher and have helped several children through this. It takes time and lots of patience! Ask your dd to cup her hands and hold water in them. Can she bring the water in her hands to her face and blow bubbles in it? Wash her face with the water? Encourage her to blow bubbles whenever her face / nose is in water, whether its running down her face in the shower or in cupped hands. I suspect she may have had water up her nose before, which won't help her reticence to put her face in water, but blowing bubbles will prevent this happening. Try to get her in the bubble habit.
Once she can put her face in cupped hands of water, see if she can do the same in an ice cream tub of water (starting small, can you see where we're going with this?!) Get her to blow bubbles. See if she can just touch the tip of her nose to the water. Can she put just her chin in? With small children, I get them to imagine it's their favourite ice cream.... Go bit by bit. Can she put in her forehead and keep her nose dry? Eventually she will be able to put her face in. Let her hold her nose if she wants to but see if she can manage without.
She will get there. In the pool, let her sit in a very shallow part of the pool, facing the side or corner and put in her chin, then the tip of her nose etc. Reduce the size of the body of water her face is entering as much as possible. Facing a huge pool and putting your face in the water is much more difficult than an ice cream tub or the corner of the shallow end. Lots of congratulations and cheering throughout btw, every time. Make a huge fuss of her doing anything that involves part of her head or face touching the water.
Spend 5 minutes each lesson on this, don't let it be the focus of the lesson, do other swimming too and let her know how well she's doing in the rest of the lesson. If she gets upset or frustrated, change the focus to another skill that she enjoys and is good at, then come back to it. This is 100% a confidence thing, you need to get that confidence sky high and she'll crack this.
Good luck and let us know how you get on.