Yes, it is fine to give a Chinchilla bananas to eat, but make sure you give them a variety of foods and not only bananas.
The fauna of South America consists of a huge variety of unique animals some of which evolved in relative isolation. The isolation of South America had an abrupt end some few million years ago when the Isthmus of Panama was formed allowing small scale migration of animals that would result in the Great American Interchange.
Some examples of animals in South America appear below: Chinchillidae
Food is any substance consumed to provide nutritional support for the body. It is usually of plant or animal origin, and contains essential nutrients, such as carbohydrates, fats, proteins, vitamins, or minerals. The substance is ingested by an organism and assimilated by the organism's cells in an effort to produce energy, maintain life, or stimulate growth.
Historically, people secured food through two methods: hunting and gathering, and agriculture. Today, most of the food energy consumed by the world population is supplied by the food industry. Chinchilla