Rest of the reindeer ate mother's last lime off (9)
I believe the answer is:
remainder
'rest' is the definition.
(synonyms)
'last lime off' is the wordplay.
'last' becomes 'remain' (I've seen this before).
'lime off' becomes 'der' (I can't justify this - if you can you should give a lot more credence to this answer).
'remain'+'der'='REMAINDER'
'of the reindeer ate mother's' is the link.
I am not very happy about this link. Some or all of it may be part of another bit of the clue.
Can you help me to learn more?
(Other definitions for remainder that I've seen before include "Rest - sell off (books) cheap" , "butt" , "Unsold book; rest" , "Dispose of unsold copies of book at a reduced price" , "Dispose of unsold books cheaply" .)