Yes, this is prey drive. If you're not well versed in training for what's known as redirection, then I suggest you find a good trainer in your area that specializes in bully breeds to assist you in this. It's a dog's natural instinct to go after small prey like you listed. You and your dog need to be trained to redirect attention and get her under control.
You didn't lose your girl, you just lost your turn!
"In this breed, there's no such thing as an expert! The day someone says they're an expert, they need to walk away from the dogs, because someone's going to get hurt. You should be learning something new every day." John Maynard-Circle M Kennels