In any non-US islands (which means, anything other than Puerto Rico
and the USVI) you would have to get a long-term visa. The conditions for getting that vary from island to island; some being relatively easy, and others being pretty complicated and/or expensive.
Probably best to focus on the US territories first, and see if you can find what you are looking for there. If not, then start looking at other island nations, and begin the process of researching visa requirements.