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.
I believe the label of “witch animal” for kitsune might come from U. A Casal’s 1959 article “The Goblin Fox and Badger and Other Witch Animals of Japan.”* There are plenty of problematic terms plus notable biases in the text, but it does contain a good collection of kitsune tales. Casal’s description of “fox-owners” or kitsune-mochi, who often do use magical means to make a kitsune do their bidding, does make it sound like a witch and familiar relationship. In that case, however, the entire family is generally connected to the kitsune (often a whole “pack” of them) and said family risked harm by offending their own foxes as well as stigmatization by neighbors. This is just one of many types of kitsune and does not make them witch animals any more than dogs would be witch animals because of the occasional inugami, though Casal does have a section on dogs for that reason.
As to whether kitsune can die or be killed, the answer is yes. And lots of people wanted to kill them. It’s noted to be a dangerous act, particularly for powerful foxes, though Casal says if “an intrepid man finds a fox asleep, he will unfailingly shoot him in all the folktales (Casal 1959, 260).” It's not clear if this passage is referring to kitsune with actual powers or random normal foxes, but even really powerful foxes could be killed by relatively ordinary weapons.
Probably the most famous story of a kitsune being killed is Tamamo no Mae. She’s often said to be over a thousand years old and have nine tails by her death, though images in this picture scroll depict her with only two. Disguised as a beautiful woman, she became a consort of Emperor Toba (1103 – 1156) and nearly killed him by draining his energy until she was revealed by a diviner and hunted down by archers. Her body was taken to the capital, though either the body or her spirit later became housed by a stone known as the Sesshō seki.
Tamamo no Mae took a long hunt by skilled archers to kill her, but many kitsune seem to die rather easily even if they have supernatural powers. In Kitsune: Japan's Fox of Mystery, Romance and Humor, Kiyosho Nozaki shares a story of a samurai who falls for what appears to be an 18-year-old girl. He asks her to profess her love for him and continues to pressure her even when she tells him she will die if she does. She finally relents and after an intimate night, asks for his fan as a keepsake. Later an old woman tells him her daughter is dead and takes him to see “a young fox lying dead on the floor, its face covered with an open white fan, the very fan given by him!” The same chapter has a story of a kitsune that takes the form of a cedar tree only to die when a Shinto priest’s son and his servant shoot arrows into the trunk. The tree immediately disappears and they find an “old fox” dead where it stood.
I think modern reinventions of kitsune have given it an increasingly invulnerable image that may not be true to the original legends. For instance, I’ve frequently seen the claim on modern sites--without citations--that the “only” way to kill a kitsune is to cut off its original tail or all its tails. I have yet to find any source backing up this claim. That said, it is an alternative to separating the kitsune from their star ball.
*I accessed this article through JSTOR through my library. You may need to search for a PDF version if you don't have JSTOR access.
