No! A physically normal 7 year old can pedal every bit as fast as it's appropriate for a 7 year old to go. The exercise will be good for her, she won't mind it, and she won't speed too much beyond her riding ability.
If you want to splash out on a nice bike, give her a 3-speed internal gear and a handbrake or two. A basket and dynamo lights would be nice kid-friendly features.