The most common and most likely answer to this crossword clue is the 8 letter word EWE-LAMB
I've seen this crossword clue in The Guardian
E
W
E
-
L
A
M
B
More clues leading to the result EWE-LAMB
Buddy explains!

The straight part
We know that "dearest possession" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case dearest possession = ewe-lamb.The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.