It's not true about a donkey being a dog! (6)
I believe the answer is:
lassie
'a dog' is the definition.
(I know that Lassie is a fictional dog)
'it's not true about a donkey' is the wordplay.
'it's not true' becomes 'lie' (I've seen this in another clue).
'about' means one lot of letters goes inside another.
'a donkey' becomes 'ass' (ass is a term for a donkey).
'lie' enclosing 'ass' is 'LASSIE'.
'being' acts as a link.
(Other definitions for lassie that I've seen before include "Scots girl - film dog" , "Screen canine" , "Girl from Scotland and dog from film" , "TV series/film dog" , "Canine film star" .)