An unidentified number of attackers killed a 65-year-old man in the north-eastern village of Bonpara.
The area is one of the country's oldest Christian communities and Islamic State have claimed responsibility for the attack.
Shafiqul Islam, deputy police chief of Natore district, said: "Sunhil Gomes was hacked to death at his grocery store near a church at Bonpara village."
The killing was part of a series of operations by Islamic State, according to the Search for International Terrorist Entities monitoring group.
Sunhil Gomes was killed hours after the partner of a anti-terror officer was murdered.
Fr Bikash Hubert Reberio, from Bonpara Catholic Church, said that Mr Gomes was "known for his humility".
He went on: "I can't imagine how anyone can kill such an innocent man."
A spokesperson for Bangladeshi authorities said the motive for killing Mr Gomes is unclear at present.
There has been a spate of both religious and secular killings in Bangladesh in recent years, with more than 40 people killed in the last three years.
