The most common and most likely answer to this crossword clue is the 9 plus 2 letter word DEEP-SEAT ED
I've seen this crossword clue in The Guardian
D
E
E
P
-
S
E
A
T
E
D
Buddy explains!
The straight part
We know that "Well-established" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case well-established = deep-seat ed.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.