The FCA's Supervision Manual (SUP) addresses the day to day relationship between the FCA, authorized persons (firms), key individuals within them, their appointed representatives and tied agents, and those who own or control them. SUP applies not only to UK firms (including, in part, to business they undertake outside the UK), but also to certain non-UK firms,specifically those "passporting" in their business from elsewhere in the EU (in co-operation with the relevant overseas regulator).
There are a number of recordkeeping requirements in SUP, of which the key ones pertain to records required by actuaries, records of appointed representatives and tied agents, persistency reports and data reports, and data relating to the use of the "passport" under relevant EU laws. SUP provides a compiled list of all of these requirements, and the associated retention periods, in Schedule 1.
Link to FCA's table of recordkeeping requirements found in SUP: https://www.handbook.fca.org.uk/handbook/SUP/Sch/1.pdf
Link to SUP Handbook: https://www.handbook.fca.org.uk/handbook/SUP.pdf