The most common and most likely answer to this crossword clue is the 6 letter word MORTAL
I've seen this crossword clue in Financial Times
M
O
R
T
A
L
More clues leading to the result MORTAL
- Lout finally collared by virtuous man
- Human’s memory recalled most of story
- Human being virtuous? About time
- Human ethical argument finally accepted
- Being completely out of it
- Socrates, for example, put little time into lesson
- Human right to treasure little time
- Person has time during lesson
- Source of tension in lesson is human
- Killing time, just about
Buddy explains!

The straight part
We know that "Killing" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case killing = mortal.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 MORTAL mean?
MORTAL noun- A human; someone susceptible to death.
- Susceptible to death by aging, sickness, injury, or wound; not immortal.
- Causing death; deadly, fatal, killing, lethal (now only of wounds, injuries etc.).
- Punishable by death.
- Fatally vulnerable.
- Of or relating to the time of death.
- Affecting as if with power to kill; deathly.
- Human; belonging or pertaining to people who are mortal.
- Very painful or tedious; wearisome.
- Very drunk; wasted; smashed.
- Of a sin: involving the penalty of spiritual death, rather than merely venial.
- Mortally; enough to cause death.