she's very cute!
I had a similar problem with my mare (recently had to give her up, not for these reasons though) but I was told to either use a dually halter or lope the lead rein over her nose as it adds more pressure for them to run though making it difficult.
I only had this problem with my mare when going back down to the field to turn her out, she would start prancing on the spot & trotting sideways just wanting to take off. I was told to give her a loose rein but as soon as her shoulders go past mine, to make her back up at least 10 steps and quickly. She would also resort to rearing if she was made to stop but I could always tell when she was about to do it so before she got a chance I turned her in a tight circle, then continued on. My mare would also start swinging her head about whenever she was asked to stop, I got headbutted a few times haha but that shouldnt be a problem with your mare since she's only little :P its very frustrating with such a hot headed horse. Think how I felt after I had to keep my mare on box rest for two weeks. Took two people to lead her anywhere :P but after persistent ground work and hours of teaching her patience, and that she'd get where she wanted to go quicker if she listened to me, we finally had a break through and she ended up being a very sweet mare that anyone could handle :)
Little tip though, when you back her up make sure you stand to the side of her instead of directly in front, I made this mistake and my mare reared & actually jumped on top of me and trampled. Wasnt pretty >_<
Oh and another thing I found worked, but you may find it difficult since your pony is so small but I used to put my elbow on her neck where it joins to her shoulder, and turn her head towards me that way she couldnt run forward. This was a last resort if I couldnt get control of her.
Try plenty of lunging & teaching her the voice commands for walk/trot/canter/woah while on the lunge & really get her listening to you. If need be let her run it out for a few minutes then once she's released some of that energy get her listening to you :) good luck!