Muse is extremely cool, with heavenly body (4)
I believe the answer is:
clio
'muse' is the definition.
(muse of history in Greek myth)
'extremely cool with heavenly body' is the wordplay.
'extremely' means to remove the middle letters (only the extremes of the word are used).
'with heavenly body' becomes 'io' (I can't explain this - if you can you should believe this answer much more).
'cool' with its centre removed is 'cl'.
'cl'+'io'='CLIO'
'is' is the link.
Can you help me to learn more?
(Other definitions for clio that I've seen before include "One of the Muses" , "Muse of history (from Renault?)" , "one historically inspiring" , "one of nine sisters" , "1 across's Muse - from Renault?" .)