I can relate to your daughter. I started riding about 3 and a half years ago and about 6 months in I was riding a friends horse. She was pressuring me to canter but I'd never done it before and was terrified, and her horse certaintly isn't bombproof, anyway I fell off twice and he stood on my leg quite hard and after that I was terrified of cantering.
When it came to cantering in my lessons I just couldn't do it, I was so afraid and did exactly what your daughter does. I never fell off having a lesson but I just didn't feel safe and this went on for about 6 months and I wanted to quit riding. I asked on here for advice and listened to what people said and with a lot of practice I eventually could canter.
It took a long time but now I love cantering, actually its my favourite gait.
One of the things which helped me was to sing a song, because when I cantered I shouted stop to whoever was running next to me and then they would stop the horse for me, so by singing It made me stop saying stop. It made me breathe because I used to stop breathing and tense up so by singing I would breath and then I relaxed and the horse relaxed.
Time: It took me a good 6 months (maybe more) to canter comfortably, I just had to practice loads (hard when you only have 1 lesson a week) but eventually I got it.
The best feeling ever was when I did my first canter on my own and sat to it and never panicked, I'll never forget that moment.
So my advice is to tell her to sing when she canters, I would suggest going on the lunge but I never did (apart from the time with my friend) and reassure her, It takes time but don't rush her and pressure her. Let her do it when she's confortable but also don't let her shy away from it forever. I can guarantee in a few months time she'll be cantering round like a pro!