can i apply for apple pay outside of the US like in the carrbean

You don't apply for Apple Pay at all. Apple Pay is something you simply use where it is accepted.

You can use Apple Pay, wherever it's accepted. There are a number of countries that support Apple Pay, so you can use it to pay in stores, restaurants, locations that accept it within those countries. Countries and regions that support Apple Pay - Apple Support

Apple Pay uses your own credit or debit cards to pay. Link-> Make purchases using Apple Pay - Apple Support

If you mean Apple Cash (different to Apple Pay), then currently it's only supported for U.S Citizens and residents over the age of 18. However, any money in the Apple Cash account, can be used anywhere Apple Pay is accepted even outside the U.S.

Link-> https://applecash.greendot.com/termsconditions/

D. Foreign Transactions.

Once you have registered your Apple Cash Account, you may use your Apple Cash Card for transactions outside the U.S. at eligible merchant locations that accept both Apple Pay and the payment network on which your Apple Cash Card is issued (please see the subsection entitled “Registering your Apple Cash Account with Bank” in Section 1(F) above for more information on this process). If you conduct a purchase transaction in a currency other than U.S. dollars, the merchant, the payment network for the Apple Cash Card, or other association that processes that transaction may convert any related debit or credit into U.S. dollars in accordance with their then current policies. For example, the conversion rate may be either: (i) selected from a range of rates available in the currency markets (which may be different from the rate the merchant, the payment network for the Apple Cash Card, or other association involved in processing your transaction is actually charged); or (ii) a government-mandated rate. The conversion rate may be different from the rate in effect on the date of your transaction and the date it is posted to your Apple Cash Card.