It's nice you tied this into the year of the rabbit. But the saying is "deer caught in the headlights." I'm not sure where it originated, but in the U.S. we say that because native deer will freeze in the middle of the road when the shine of the headlights hits them. This is different from most other animals that will run when they see a car coming. In fact, there are places in Appalachia at least where people take old trucks, and a spotlight, and will purposefully shine the spotlight on the deer in order to hunt them for food and sport. Sounds barbaric, but just explaining the saying and some reasons why we say "deer" as opposed to another animal.

Never heard this with a rabbit. I've always said deer.

An interesting expression! In the States, we say like a deer caught in the headlights.