The officer is now in custody, according to the report.
The police officer fired his weapon after a verbal argument with the man selling the counterfeit product in an open-air market in Kisii County, western Kenya, on Sunday, the report said.
After the shooting, protesters quickly descended on the police station, throwing rocks and setting bonfires, damaging the station and nearby vehicles.
Fourteen police officers were injured and protesters dispersed after an "unquantified number of live and blank ammunitions were used", the report said.
Accusations of police brutality are common in Kenya. Since the coronavirus outbreak, multiple cases of alleged police violence towards lockdown violators have been reported.
In March, in the first 10 days of a curfew to curb the coronavirus, at least six people were killed, according to rights group Human Rights Watch.