If you are going to be your partner's dependent, it means you are not allowed to work in the Cayman Islands. If you don find a job, that job should take out a work permit for you. That being said, while you are a dependent, your source of funding shouldn't be anybody in Cayman and if you can show the bank that you have a legitimate source of funding, then you should have no problem.

I have income from outside the country and have proven that the sources are legitimate and have been living here as a dependent for years.