"The toll now stands at eight as two more injured succumbed to their wounds," a police officer told reporters hours after a group of at least 15 robbers raided the Bangladesh Commerce Bank's Kathgara branch.
He said that people in the neighbourhood retaliated, lynching one of the robbers and injuring two others as they tried to flee the scene.
Police recovered Taka 600,000 recovered from the captured dacoits.
Witnesses said other robbers fled the scene on motorbikes.
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Police said it is not clear how much money was taken away from the bank.
Two of the deceased have been identified as the branch's manager Md Oliullah and a customer.
At least 8 people were injured during gunfire and they are being treated at a nearby hospital.
The bank officials said police were examining the bank's CCTV footage to identify the robbers.