C.B.
As a swimming instructor who was "always" given the kids and adults who were afraid of water.....
You cannot allow her to quit. As a child, she needs a little more one-on-one instruction or better yet, would be a Mommy and Me class, where YOU learn the techniques to help get her into the pool, and blowing bubbles.
My first question to all who came to my classes, "Do you know what caused you to be afraid of the water?" The answers always varied from, "I don't know" to "When I was...I almost drowned..." type of things.
My first lesson was always, just to sit on the steps of the pool, up to the chest, and spend the 30 - 45 minutes just talking, and then pointing out that they spent the time in the pool and nothing bad happened.
The next day, we would sit in the water, and blow bubbles and walk across the shallow end with kick boards and blowing bubbles being a "boat". That would then go to bouncing, up and down, so that you are completely submerging (but when you stand, the water is only at your chest). I would also introduce floating on the back, letting everyone know that if they get into trouble or get panicked, they can float on their back and breathe.
The next day we would "warm up" with the bouncing, blowing bubbles and floating. Then we would use the kick boards and start to kick across the pool (no threat, they don't have to put their face in the water). Then after a few times across, I would have them blow bubbles and be a motor boat. A few more times, and I would have them put their face in the water as they go across the pool. Then I would have something of interest for the students. I would show it to them (for the adults, I had a silver dollar), and tell them it was theirs if they could retrieve it from the bottom of the pool. I then put the object in a sinking object and tossed it in to just above their shoulders deep water (they had to get the object with their HANDS not their feet). Normally this got them very used to having their face in the water and realizing, THEY are in control.
I constantly worked on going from tummy to back (back floating) and reminding my students that anytime they are tired, or scared, to float on their back, and BREATHE. I also had exercises where they followed the bubbles...UP. That way, if they found themselves in "deep" water and got confused, they would remember "follow the bubbles" and would get to the surface where they could back float.
It is really wonderful to watch a scared person realize they CAN swim and are in control of the water, and go off the diving board when only two weeks before they had a hard time even getting into the pool.