The suspect, in his late 70s, fled the scene in his car after the incident in Holon, southeast of Tel Aviv, but was later arrested, police said.
The nurse who died was said to be a woman around 40 years old.
A senior police officer at the scene told news site Ynet that "a man went to receive medical treatment at the clinic here in Holon.
The director of the Clalit health service that operates the clinic said in an interview with public radio that the perpetrator's "medical background" was known, implying the patient might have had mental issues.
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