Duterte was sworn into office on Thursday after a landslide win built largely on a pledge to kill thousands of suspected criminals in a relentless law-and-order crackdown he said was needed to stop the Philippines becoming a narco-state.
Police in Bulacan province neighbouring Manila said they launched an anti-drug trafficking blitz on Thursday, the day Duterte took office, that claimed nine lives.
"We are just following the marching orders of our new chief of the Philippine National Police," Bulacan police spokeswoman Senior Inspector Linel Solomon said.
Another drug suspect was also shot dead after he used a gun during a raid on his home in the central city of Dumaguete on yesterday, local police desk officer Mark Tikling told AFP.
With Duterte encouraging them, police had already killed dozens of suspected criminals between his election victory and being sworn into office.
Duterte, who has been accused of links to vigilante death squads that killed more than 1,000 people during his nearly two decades as mayor of southern Davao city, insisted during his inauguration speech that he would not break the law during his war on crime.
He also promised to use his presidential powers to protect police who faced legal charges for mass killings.
"If in the process you kill 1,000 persons because you were doing your duty, then I will protect you," he said in a speech to install the country's new chief of police, Ronald Dela Rosa.
Dela Rosa also warned yesterday that police officers linked to drug trafficking to surrender by tomorrow or be killed.
Spokesmen for the new president could not be contacted for comment on the new killings.