I wouldn't push it if it's scaring her that much. I would just teach her myself at her own pace. Let her take it slow with what is comfortable with her. My daughter just turned 5 & she hasn't learned to swim yet either. I'm hoping to make some progress this year!
Last year she decided on her own to try to put her face in the water. "Look! Watch me Mommy!" She started out, holding onto the ladder, and just put her mouth underwater for a second & was so proud! Then she held her nose & got her mouth and nose under water. I encouraged her, did it with her, and told her how proud I was. A couple time she made it to holding her nose, closing her eyes & getting her whole face under. I did not push her to do any of it, I tell her to do what she is comfortable with.
Oh, forgot to add that my brother, sister & I took lessons together when we were young. HORRIBLE experience! The lady was awful and all I learned to do was hold the ladder & blow bubbles in the water... lol Dad ended up teaching me later to "doggy paddle". I'm not an expert swimmer today, but I won't drown!