The most common and most likely answer to this crossword clue is the 5 plus 6 letter word RIGOR MORTIS
I've seen this crossword clue in The Times
R
I
G
O
R
M
O
R
T
I
S
More clues leading to the result RIGOR MORTIS
Buddy explains!

The straight part
We know that "no signs of breathing here" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case no signs of breathing here = rigor mortis.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.