Ship might sink following drills (9)
I believe the answer is:
mayflower
'ship' is the definition.
(I have seen 'old ship' mean 'mayflower' so perhaps 'ship' could also mean 'mayflower')
'might sink following drills' is the wordplay.
'might' becomes 'may' (I've seen this before).
'sink' becomes 'lower' (I've seen this in another clue).
'following' becomes 'f' (used when citing page numbers etc).
'drills' indicates putting letters inside (to drill can mean to make a hole in something).
'may'+'lower'='maylower'
'maylower' going around 'f' is 'MAYFLOWER'.
(Other definitions for mayflower that I've seen before include "emigrants' vessel?" , "vessel crossing ocean" , "Historic vessel" , "Hawthorne mentioned" , "Pilgrim Fathers' vessel" .)