Unfortunately, Chinchillas are banned pets in Australia. Along with other pets that they classify as exotic, they believe that they are a threat to the natural ecosystem; thus, it is illegal. Other pets that Australia classify as exotic and illegal are gerbils and hamsters. That is not the same as the Guinea pig, as they are not seen as a threat to the ecosystem because they were domesticated in Australia before other animals were seen as threats.
Chinchillas are thought by some to be able to live and reproduce in the wild in Australia. The Australians are unsure how the introduction of them into their ecosystem would change the balance. However, other people believe that they would be unable to livepast a single generation in the wild. That is because they are susceptible creatures who would struggle with bedding, temperatures, and food.
However, whichever is true, it is illegal to own a chinchilla in Australia.