The most common and most likely answer to this crossword clue is the 11 letter word PRINCELIEST
I've seen this crossword clue in The Guardian
P
R
I
N
C
E
L
I
E
S
T
Buddy explains!

The straight part
We know that "Extre/m/ely lavish" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case extre/m/ely lavish = princeliest.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.
What does PRINCELIEST mean?
PRINCELIEST adjective- Relating to a prince; regal; royal.
- Befitting a prince; grand; lavish or opulent.