No, this is not possible.
PostgreSQL is a relational DBMS, operating most efficiently on properly normalized data models. Arrays are not relational data structures - by definition they are sets - and while the SQL standard supports defining foreign keys on array elements, PostgreSQL currently does not support it. There is an (dormant? no activity on commitfest since February 2021) effort to implement this - see this answer to this same question - so the functionality might one day be supported.
For the time being you can, however, build a perfectly fine database with array elements linking to primary keys in other tables. Those array elements, however, can not be declared to be foreign keys and the DBMS will therefore not maintain referential integrity. Using an appropriate set of triggers (both on the referenced and referencing tables, as a change in either would have to trigger a check and possible update on the other) one would in principle be able to implement referential integrity over the array elements but the performance is unlikely to be stellar (because indexes would not be used, for instance).