Athanassios Arvanitis, a 31-year-old chef, was shot five times by police before his arrest. He has wounds to his lungs and is in intensive care.
The authorities believe the deadly rampage was triggered by his dismissal as a chef in a hotel.
Three women injured during the police chase are said to be out of danger.
The trail of violence began when the man used a butcher’s knife to decapitate his girlfriend’s dog on the balcony of their home in the village of Vourvoulos, near Santorini’s main town Thira.
Next, Mr Arvanitis allegedly killed his girlfriend, a 25-year-old teacher called Adamantia Karkali, by stabbing her in the armpit.
He then cut her head off and began parading it around the streets.
During a dramatic chase, Mr Arvanitis tried and failed to behead a policeman. He was then shot five times and arrested.
Doctors at Santorini’s medical centre say that despite his injuries they found it almost impossible to restrain him.
Two women doctors knocked from their motorbike by Mr Arvanitis as he tried to get away in a police jeep are said to be out of danger, as is a Greek woman tourist who was hit by a ricocheting bullet as she watched the sunset.
People living in Santorini say they have not witnessed such savagery on this iconically beautiful island since the Germans carried out executions during the World War II occupation.
By Malcolm Brabant , BBC News, Athens
