Chinchillas are Prone to Heatstroke
Chinchillas have fur for a reason. In the wild they live in the Andes mountains where it gets cold. Their fur is what keeps them warm. However, domestic chinchillas are prone to over-heating since most people's homes are much warmer than the Andes.
The ambient temperature where chinchillas are kept shouldn't rise above 80 degrees. But temperature is not the only consideration as humidity also plays a role. A good rule of thumb is to add the temperature and the relative humidity together. If this number exceeds 150, you should take action to reduce both.
Failing to maintain a proper temperature can result in heatstroke. This is a dangerous situation and one that can be fatal. Warning signs include panting and lying down stretched out on one side.
If your chinchilla is suffering from heatstroke you should seek the attention of a vet. However, to bring the chinchilla's body temperature down consider offering some cool water to drink, wrapping a cold wet towel around the body, or submerging the chinchilla in cool (not freezing) water. As you undoubtedly know, getting a chinchilla wet is generally a bad idea, but heatstroke is a particularly dangerous situation that needs immediate attention.
Bettina Hansen's (website no longer live)
A New Owner's Guide to Chinchillas by Audrey Pavia