Dig west half of island (4)

I believe the answer is:
barb
'dig west' is the definition.
'barb' can be an answer for 'dig' (I've seen this in another clue). I'm not certain of the 'west' bit.
'half of island' is the wordplay.
'half of' means to take half the letters.
'island' becomes 'Barbados' (island nation).
'barbados' halved is 'BARB'.
Can you help me to learn more?
(Other definitions for barb that I've seen before include "Arrowhead projection" , "Kind of thorn or wounding remark" , "Aggressive remark intended to have a telling effect" , "Cutting remark" , "Jibe" .)
