The most common and most likely answer to this crossword clue is the 9 letter word ARROGANCE
I've seen this crossword clue in Independent
A
R
R
O
G
A
N
C
E
More clues leading to the result ARROGANCE
- Superior approach in developing organ care
- Cargo near wrong side?
- Novel orange car conveying self-importance?
- Pride of farm crop, egg was once set to be shelled
- Pride shown by Josh’s leader in mixed race
- A classic car with which old French director shows conceit
- Boasting of exotic orange car
- Conceit from playing organ in new care home?
- Turn up in Scottish island church showing disdain
- Bluster and rage on car breaking down
Buddy explains!

The straight part
We know that "conceit" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case conceit = arrogance.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 ARROGANCE mean?
ARROGANCE noun- The state of being arrogant; a type of extreme or foolish pride in which someone feels much superior to another