Authorities said the dead suspects and three others on the run were stragglers from a boatload of southern Philippines based Abu Sayyaf gunmen who sailed to the central island of Bohol this month as part of a plot to kidnap tourists.
The raid had signalled an escalation of the kidnapping threat from the Abu Sayyaf, who are based on remote islands and also blamed for beheadings and deadly bombings, prompting Duterte to vow deadly retribution.
"I told the military: Do not sleep. Find them. Kill them. I want them dead. Do not give them to me alive," Duterte said about the fresh fighting in Bohol.
"They are animals but if you want me to be an animal, I am used to that. We are the same. I can dish out more than what you can," he told people attending a public sports event.
Six other Abu Sayyaf members had been killed in clashes on the island on April 11, when three soldiers and a policeman were also slain, authorities said.
The military said they had been pursuing up to seven of the remaining gunmen.
Security forces continued the hunt for the remaining three gunmen on Sunday while residents fled to safety to avoid getting caught in the crossfire.
"The remaining lawless armed elements who are strangers in the area have nowhere to go," a joint military and police statement said.
"It could just be a matter of time before we can say that the threat (on Bohol) is totally eliminated."
The Abu Sayyaf, made up of several hundred terrorists, is notorious for kidnappings, beheadings and deadly bombings. Its leaders have also pledged allegiance to ISIS fighters in Iraq and Syria.
On Sunday, the army said the Abu Sayyaf beheaded a soldier it abducted last week in its southern Jolo stronghold.
The fighting in Bohol, about 500 kilometres (300 miles) away from Jolo and 800 kilometres (500 miles) south of Manila, caused a scare for the Philippines' important tourism industry.
The Philippines alerted Western governments after the military said it got wind of an Abu Sayyaf plan to raid central Philippine resorts during the Easter holidays and kidnap up to a dozen tourists.
This led to travel warnings being issued by the United States, Australia, and other countries.
Over the past year the Abu Sayyaf has been expanding its activities from the south where the military has been waging an offensive since last year.
Its gunmen have been boarding commercial and fishing vessels and abducting dozens of foreign crew members, ransoming some of them off for huge sums of money, according to the defence ministry.