The gunmen struck the town of Ringim late on Thursday night, said Abdul Jinjiri, police spokesman for Jigawa state, where the town is located.
"After attacking a police station and killing three of our men, they then moved to nearby Unity Bank and killed two civilians there," Jinjiri said. "They were not able to steal any money from the bank."
The Ringim attack came a day before 25 people were killed in a separate clash between Nigerian security forces and suspected Islamists who robbed a bank and attacked a police station in northeastern Yobe state.
Authorities believe Islamist are increasingly targeting banks to fund their operations, as support from local politicians dries up and Sahara-based groups are in disarray after a French military intervention kicked them out of Mali.
Jinjiri declined to say if the Islamist group Boko Haram, which operates across the north, was suspected of being behind Thursday's clashes.
The Boko Haram sect and offshoots such as the al Qaeda-linked Ansaru, as well as associated criminal networks, pose the main threat to stability in Africa's top energy producer.
Violence in the country's north has shown no signs of letting up. Clashes between Islamists and a multinational force from Nigeria, Niger and Chad killed dozens of people ten days ago.
A senator who visited the site said 228 people were killed, but the military puts the figure at 37.
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