Billy Elliott carries the caustic solution (3)
I believe the answer is:
lye
'caustic solution' is the definition.
(I've seen this in another clue)
'billy elliott carries' is the wordplay.
'carries' indicates the answer is hidden within the clue.
'LYE' is hidden within 'billy elliott'.
'the' is the link.
(Other definitions for lye that I've seen before include "Alkaline liquid" , "Strongly alkaline wash" , "caustic liquid" , "solution that comes out in the wash?" , "Strongly alkaline solution for washing or cleansing" .)