Our valued sponsor

Bank account for a Cayman Islans exempt company

You may look into some of the Cyprus banks or if you have lot's of money and are willing to travel you may go with some of the swiss banks for instant CIM Bank!
 
I don't think Cayman Island is on CIM's list of approved countries they open accounts for last time I checked. The only offshore company they allowed was a RAK company and I think an Isle of Man company. May have changed however.

The US part is an issue. It will greatly limit your options with smaller banks. What activity and where the activity will be taking place will also affect which banks would consider you.