The Carrot Pen is, as the name implies, a pen shaped like a carrot that belongs to ZPD Officer Judy Hopps. Besides being a pen, it also functions as a voice recorder, which benefits Judy whenever she needs audio evidence in her assignments.
Role in the film
Judy uses the pen to record precise dialogue from Nick Wilde that constituted a confession on his part that made him guilty of tax evasion, and thus blackmails Nick and will only give it to him only if he helps her find a missing otter named Mr. Otterton.
After going to the Mystic Springs Oasis and the Department of Mammal Vehicles, Nick wants the pen but Judy wants to go to where the limo was. At the limo service, Judy offers Nick the pen but throws it over the fence, leaving him no choice but to climb the fence to get it. But upon doing so, he finds Judy on the other side, holding the pen, having dug under the fence, and explained, while she couldn't enter the premises without a warrant, she had probable cause by seeing a "shifty lowlife climbing the fence", giving her more dirt to blackmail Nick with, as well as enabling her access to the premises.
Later, when the case was solved, Judy gives Nick the pen, and by extension the blackmailing recording, as a sign of the deal fulfilled and Judy accepting him. She also suggests he uses it to fill in an application for the ZPD.
After Judy figures out what's causing the predators to go savage and finding Nick, he records Judy's saying "I really am just a dumb bunny!" after admitting her mistake, and plays it back twice. When being comforted, Nick playfully asks if she's doing this to get the pen for recording the last thing she said until he tells her she's standing on his tail.
The pen is used again to record the monologue of Bellwether as she admits being the mastermind behind the predators going savage, which Judy and Nick play back right as the ZPD arrive. With this evidence, Bellwether is put behind bars and the predators are reunited with the prey.