A few years back, I had a dog who died of overheating. It was pretty surreal; I'm taking the dog for a short walk on a mild Spring day, he starts breathing in the most fucked up way, and an hour later the vet tells me that there's really nothing that can be done, that his internal temperature is high enough to cause irreparable brain damage. Within two hours of the first symptoms, my dog is dead. That really fucking sucked. He had cancer, so I guess something just malfunctioned.
Hope it works out for you. Typically, you'd want to wrap the pet in lukewarm towels, and provide some form of breeze to slowly chill the towels down. Otherwise, supposedly it could backfire the other way, and the pet could go into a coma. At least that's what the vet told us. No idea how things work with chinchillas.