Island dispatches its leader in a very cold fashion (5)
I believe the answer is:
icily
'island dispatches its leader' is the definition.
I can't tell whether this definition defines the answer.
'in a very cold fashion' is the wordplay.
'in' indicates putting letters inside.
'a very cold' becomes 'icy' (I've seen this in another clue).
'fashion' becomes 'il' (I can't explain this - if you can you should give a lot more credence to this answer).
'icy' enclosing 'il' is 'ICILY'.
Can you help me to learn more?
(Other definitions for icily that I've seen before include "Without feeling" , "In a cool manner" , "Coldly" , "In a frosty manner" , "In a cold manner" .)