Merseyside Fire and Rescue Service said 12 engines were at the scene at King's Dock. Merseyside Police said a number of vehicles were on fire.
Nearby apartments have been evacuated due to smoke billowing from the blaze, the force added.
Merseyside Police has warned people to stay away from the area, adding that travel restrictions were in place.
It is understood there are about 150 cars in the car park and people are being told to leave their vehicles there for the night.
The Liverpool International Horse Show has been running at the arena, which has a total capacity of 11,000, since 28 December. On the show's Facebook page, organisers said it was with "considerable regret" that it had decided to cancel the evening show due to the fire.
It added: "All people and horses are safe and secure, and show organisers have thanked spectators, riders and support teams for their understanding and co-operation during this ongoing situation.