Terrified residents of the popular tourist island barricaded themselves into their homes and called the police.
The suspect, 35, was shot during a dramatic car chase in which he crashed into a motorbike and badly injured the rider and pillion passenger.
The crime is one of the most gruesome in Greece in recent memory.
Speaking on Greek television, the local sub-prefect said the man attacked officers who tried to arrest him and slashed one policeman in the face.
The man then stole a police jeep and tried to get away.
But after 400m the suspect slammed into a motorbike carrying two female doctors.
They were thrown into the air and badly hurt.
The police then opened fire and hit the alleged knifeman five times.
According to one eyewitness, one police bullet ricocheted off the road and hit a woman in the leg and jaw.
The suspect is said to have a history of jealousy and domestic violence.
The authorities say they are organising a military aircraft to fly the injured for emergency surgery in Athens.