The killings began on Saturday night and included a raid on a bar.
Witnesses say that in some cases the man demanded mobile phones and money, on other occasions he did not take anything from his victims, Tanzania’s Citizen newspaper reports.
The attacks have created fear in the district of Tarime, it says.
According to Tanzania’s Habari Leo paper, the gunman has also been attacking people he meets on the road.
Many businesses are opening later in the morning and closing earlier to avoid the gunman, and the town of Tarime resembles a ghost town at night, the Citizen reports.
Tarime police commander Justus Kamugisha told the BBC that the manhunt was still continuing five days after the attacks began.
Police investigators have arrived from Tanzania’s main city of Dar es Salaam to assist in the search, he told BBC Swahili.
The weapon the man had used was thought to be a submachine gun or an assault rifle, he said.
A special operation involving about 160 police officers was launched on Thursday morning, the police commander said.
“We will not rest until we get him and bring him to book,” Mr Kamugisha told the Citizen.
BBC Swahili TV’s Salim Kikeke says it is not unusual for people in the Mara region, which borders Lake Victoria and Kenya, to illegally carry arms in order to protect their livestock.