I absolutely won’t say this about another’s autobiography? (3,2,4,4)
I believe the answer is:
not on your life
'i absolutely won't' is the definition.
(phrase used to strongly reject an offer)
'this about another's autobiography?' is the wordplay.
I cannot really see how this works, but
'about' could be 'on' (as in a book 'on' a certain topic) and 'on' is found in the answer.
'autobiography?' could be 'life' (autobiography is a kind of life) and 'life' is found within the answer.
'another' could be 'r' (I've seen this in other clues) and 'r' is found within the answer.
This explanation may well be incorrect...
'say' acts as a link.
(Other definitions for not on your life that I've seen before include "There's no chance" , "Like someone else's autobiography" , "Never! (colloq.)" , "Emphatically, no!" , "On no account whatsoever" .)