Yes, you're being too pushy. Why is being a doctor better than being a hairdresser? Because it has more kudos? Because you earn more money? Because only an elite can do it?
All those things may be true, but, and I'd seriously be asking myself this if you really love her (and I'm sure you do), will all those things make her happy? Will she actually be happier being a doctor than a hairdresser? If she loves doing hair, I'd say, no. So why would you rather she was miserable doing something than fitted your ideal than happy doing something that fits hers?
My youngest daughter is seriously bright. Is preparing to take some GCSEs at 12 kind of bright. Was put in with a class of 11 year olds at only just 8 because the school couldn't cope kind of bright. She wants to be a baker. She could do anything, but she loves baking. So, It's baking I'm encouraging because more than anything else, I want her to have a happy life.
Also, a bright child will do well in whatever they do. So your daughter might own her own multi-million pound chain of hairdressing companies one day, if you let her go her own way.
A friend of mine is a solicitor. She went through a stage of being so depressed we were seriously worried about her. Her parents were very pushy, insisted on top university, chose her career for her. And she hates it. She once said to me "I'd have loved to have tried something ordinary, but I'd have never been allowed to, and now I'm stuck where I am." She's comfortably off, but she's miserable in comfort. I want more than that for my daughter. She's in the kitchen cooking biscuits right now