What is the name of first residential school?

There's no single answer to the question of the "first" residential school, as it depends on how you define "residential school" and which country you're focusing on. The concept of residential schools for Indigenous children arose independently in different countries and at different times, often with varying goals and levels of government involvement.

For example, in Canada, the commonly cited earliest institutions were established in the 1830s, but they weren't all run by the government and their purpose wasn't always explicitly to assimilate Indigenous children. Similarly, in the United States, Australia, and other countries with significant Indigenous populations, different institutions arose at different times with varying aims. There's no universally agreed-upon "first" among all of them.