The first in bikini, seek permission to sunbathe (4)
I believe the answer is:
bask
'to sunbathe' is the definition.
(I've seen this before)
'the first in bikini seek' is the wordplay.
'the first in bikini' becomes 'b' (1st letter of 'bikini').
'seek' becomes 'ask' (I've seen this before).
'b'+'ask'='BASK'
'permission' is the link.
I am not very happy about this link. It may be part of another bit of the clue.
Can you help me to learn more?
(Other definitions for bask that I've seen before include "Lie in sunshine" , "Soak up sun" , "revel" , "Lie in the sun" , "Sunbathe" .)