Caged gerbil, for example, given name (4)
I believe the answer is:
pent
'caged' is the definition.
(I've seen this in another clue)
'gerbil for example given name' is the wordplay.
'gerbil for example' becomes 'pet' (gerbils are often kept as pets).
'given' means one lot of letters goes inside another.
'name' becomes 'n' (common abbreviation e.g. on forms).
'pet' going around 'n' is 'PENT'.
(Other definitions for pent that I've seen before include "enclosed" , "Closely confined" .)