If you want to stay in the Netherlands for longer than 90 days, you may need a residence permit. In many cases you will also have to apply for a long-stay visa before you travel to the Netherlands. This visa is also called an authorisation for temporary stay (MVV).
It is not possible to apply for this visa in the Cayman Islands. You can submit your application at the Netherlands embassy in Washington D.C. or at the consulates-general in Miami, New York or San Francisco, United States.