Ganpapatou no Zerosen Shoujo

Asako ojou-sama told her father that she wanted to fly in the sky. One day she's taken to Ganpapa Island and receives a Zero fighter plane. And then she flies it. (Not to mention, the Zero seems to talk to her somehow.) Later she's joined in the sky by her friend Princess Kurara, who flies a P-51 Mustang. They have a mock dogfight.