When my children were small, we had family pets. THIS member of the family took care of all of them (the children helped). That was all right with me.
But when the kids wanted pets of their own, I told them that when they were out of elementary school, they could bring up the subject and we'd talk about it. It takes a high degree of maturity to commit to a living creature who needs careful, thoughtful, regular care (and that includes a bunny, a gerbil, or a snail).
So my older son eventually had some fish, and he took very good care of them. My older daughter had a series of hamsters (because, as conscientious as she was, they just didn't live very long). My younger daughter had a parakeet. My younger son decided he didn't want to do what his siblings were doing; however, now he's married and has two dogs, a cat, and two foster cats, as well as a son and a daughter, so he's making up for lost time.
It sounds to me as if your daughter may love bunnies (and so do I) but she loves getting what she wants even more. She is confident that she will wear you down if she pesters and worries you long enough. It's a form of power game. That's five-year-old strategic thinking!
You can tell her, "Abby, look at me so I know you're listening. No. Bunny. Now. Ask me again when you're twelve (or ten or whatever) if you want a bunny then. That's it. Don't. Ask. Again. Now, what did I just say?"
What if Abby starts pestering and whining right away (and you know it could happen!)? Then deal with her outright disobedience - which has nothing to do with rabbits - the way you usually do. You will not destroy her character or smash her spirit, whatever she may tell you.
This sounds mean on the surface, but it's actually a good idea NOT to give a child what she whines for. What she asks for politely is something else and you can consider it rationally, but the final decision is still up to Mama and Daddy. That's the way the world runs. Especially when you're five.