Something that might help is to just take her to a show early enough to get her comfortable in the arena before she has to be in a real class that you paid for. Also, spend some time around the entrance and let her relax, but stay out of other people's way that are going in/out. This might help the rider calm down, as she may just be picking up on her rider's nerves.
Another thing is the arena can be quite a lively place with the spectators clapping and the announcer blaring. So just going and hanging out around that noise should help her calm down. When I used to show, we would sometimes bring the green horse even if we weren't showing and just ride around the grounds and in the warm up arena to get them used to the busy environment, but without the pressure of a real class. If you live close to an arena that frequently has events, haul her to them and just ride around the chaos. A few quiet rides in and out of the gate so that it becomes "old hat" and then go home.
Also, she may be very smart and anticipate that in a real class, the gate is the end of the class,and the rider is less apt to correct her, so she tries to hustle out. Some training such as dismounting before exiting would help, as well as if you do a "schooling class" or two. Make sure the show staff are aware of what you are doing and be sure not to disrupt other horses that are competing. But you can go to the center of the arena, get off/on and stop somewhere besides the gate during a class. This gets her in the real class, but not allowed to get away with anything.
Just changing a few things now will help prevent more problems.
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