Cayman Island Records as part of Jamaica before 1962.
According to the index books for Jamaica, Cayman Island was part of the Jamaican Civil Registry until about 1962. It was designated parish P in the Codes. Are these registry books still archived in Spanish Town Jamaica or were they moved? If moved, where did they go?
The FamilySearch Wiki page https://www.familysearch.org/en/wiki/Cayman_Islands_Civil_Registration says
"For civil registration before 1962, please see Jamaica Civil Registration" This implies to me that the physical records would still be in Jamaica, but others may assume differently.
Even if records were sent to Jamaica, copies may have been kept in the Cayman Islands, so some records may be found in both places, subject to physical survival (such as hurricanes, insects etc)
The following online book published 2012 has short section on the Cayman Islands from page 151
Tracing your Caribbean ancestors : a national archives guide by Guy Grannum
Internet Archive Books to Borrow (you need to register with the IA to read more than a few pages)
One of the books suggested for further reading is described in this link, available to buy from the Society of Genealogist's Shop https://societyofgenealogists-shop.myshopify.com/products/my-ancestor-settled-british-west-indies My Ancestor Settled in the British West Indies by John Titford1