There's definitely one in Zante Town, which isn't too far from Kalamaki.

I know, because my ex-husband had to take my daughter's friend there when she fell down a ditch and we thought she might have broken her ankle. Luckily it was just a bad sprain and they bandaged it up and offered her a suppository (at least that's what he swore they said!) It was 20 years ago though, so I expect they've moved on a little since then

HOSPTAL IS FINE TOOK MY SON THERE A FAIR FEW YEARS AGO SO IMAGEN ITS EVEN BETTER NOW

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The hospital in Zahynthos is fine, as long as you don't expect it to be anything like the NHS.
Medically its good, but you must remember that in Greece all your personal care is to be provided by relatives. This includes change of bedding, all food, drinks, washing and all aspects of personal care.
It should be noted that many of the natives actually opt to go to Athens for treatment.