The closest thing I got from looking this up is a game. But I kind of need a non-modern interpretation answer for an idea I'm working on.
In the Edo period (1603–1867), kitsune were seen as witch animals. With a little research, I found out that witch in Japan were viewed more or less the same as the rest of the world (evil or with malintent). Witches used kitsune for their power to suite their own wants.
Did anyone ever wish a kitsune harm and end up killing it?
One belief is that when a kitsune changes shape, its hoshi no tama holds a portion of its magical power. Another tradition is that the pearl represents the kitsune's soul; the kitsune will die if separated from it for long. Those who obtain the ball may be able to extract a promise from the kitsune to help them in exchange for its return
This is what the Wikipedia article says about kitsune.
It also states that when kitsune gets old enough it ascends to heaven, so they are immortal in that sense.
After reaching 1,000 years of age and gaining its ninth tail, a kitsune turns a golden color, becoming a 'Tenko' ("heavenly fox"/"celestial fox"), the most powerful form of the kitsune, and then ascends to the heavens.
Kitsune is surrounded by folklore, so the stories may vary. I would suggest reading the article yourself, since It is out of scope of this answer.
But I have not found a mention about somebody killing a kitsune.
