Two civilians and a police officer died in the attack on the bank in Mandera County, police commander Charles Chacha said.
The town is located seven kilometres from the Kenya-Somalia border.
"We are seeking to establish whether it was an attempted robbery or if al Shabaab militants were involved", he said.
Shisia, however, said that the assailants had escaped by the time the police arrived.
Shisia said that he suspected that the militants were on a revenge mission after two of their top commanders were killed on Monday by Kenya Defence Forces (KDF).
AB de Villiers backs South Africa to win Test series against England
It was the highest ever score by an England player on his T20 debut, and England's 11th highest score of all time in the format. Dane Paterson , who was twice on a hat-trick, led South Africa's attack with four for 32.
He said that the militants were known to upscale attacks during the last 10 days of Ramadan and especially during dates with odd numbers. When the shots stopped, three civilians and two officers manning the bank lay dead.
Sources said there was a gunfight between the officers and the attackers and one of the terrorists was killed. This May, five police officers guarding the Mandera County governor were killed in a roadside bomb attack in Simo.
In a separate incident, the Kenya Red Cross says two people escaped unhurt after their auto ran over a homemade bomb in Mandera.
Shisia said the Improvised Explosive Device (IED) went off as the vehicle with two occupants drove near it at about 9:30 am on Friday in Lafey Sub County.
"The windscreens of the vehicle were completely shattered down and two occupants of the vehicle seriously injured due to the impact of the blast", Shisia said.
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