Sometimes — with care
Bugs Bunny lied: carrots are candy for rabbits, not a staple. The sugary root should be an occasional treat — carrot TOPS (the leafy greens) are the part rabbits should eat freely.
A rabbit’s diet should be ~85% grass hay, with leafy greens and a little pellet food. Carrot root is high in sugar by rabbit standards, and regular carrot feeding is a known contributor to obesity and dental disease in pet rabbits, plus gut flora disruption that can trigger soft stools or GI stasis. The leafy carrot tops, by contrast, are an excellent daily green. The cartoon-rabbit-with-carrot image has genuinely caused decades of well-intentioned malnutrition.
A slice or two of carrot (think thumb-sized total) a few times a week, maximum — treat it like chocolate for humans. Unlimited timothy/meadow hay and a rotating mix of leafy greens are what actually keep a rabbit healthy. Baby rabbits under 12 weeks: no carrot at all.
Carrot tops, romaine, coriander, basil and parsley make better everyday greens.
This article is general information, not veterinary advice. If your pet has eaten something potentially harmful or shows symptoms, contact your vet or an emergency clinic immediately. Full disclaimer.