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A spokesman for Afghanistan's interior ministry described those killed as "medical staff".
It was unclear whether the policeman was dead or in custody, BBC News reported.
Afghanistan suffered a spate of deadly attacks in the run-up to presidential elections held on 5 April, including a restaurant bombing in January and an attack on a hotel in March.
Thursday's attack took place at a hospital in western Kabul run by Cure, a small US Christian charity.
The US embassy in Kabul tweeted that it was "with great sadness we confirm that three Americans were killed in the attack on Cure Hospital". No other information would be released for the moment, it added.