The most common and most likely answer to this crossword clue is the 14 letter word IN-TOED/TATAMI
I
N
-
T
O
E
D
/
T
A
T
A
M
I
Buddy explains!
The straight part
We know that "Like a pigeon/mat" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case like a pigeon/mat = in-toed/tatami.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.