My MoMA rabbit has had a small growth near her nose for over a year. Started off looking like a skin tag. It would get a little larger, then shrink back down. It's quite large now and very hard like a fingernail. It does not seem to bother her or get in the way of accessing food & water. I've read a bit about what it is on the interwebs-- rabbit papilloma is my best guess- but I'd like another opinion:)
This does look typical of a cottontail rabbit papilloma. They are usually benign but can turn into squamous cell carcinoma which is malignant. I recommend you have your veterinarian remove this "horn" and submit it to a pathologist to make sure the edges are clean.