The wife of a counter-terrorism officer and a Christian man were killed in attacks carried out by suspected Islamists in Bangladesh on Sunday, police said.

Assailants on motorbike shot and killed Mahmuda Aktar Mitu, the 32-year-old wife of officer Babul Akhter, when she was on her way to drop her son off at school in the port city of Chittagong, police spokesman Paritosh Ghose told reporters.

The 6-year-old son of the police officer, who had conducted a number of raids against home-grown militants, survived the attack unhurt, he said.

"We suspect the militants are behind the murder," said Ghosh adding that Babul's investigation recently helped bust a number of dens of the banned Islamist outfit Jamaat-ul Mujahideen Bangladesh in the port city 300 kilometres south of the capital Dhaka.

Home Minister Asaduzzaman Khan said that the religious extremists might have killed Mitu when they failed to get her husband.

"Babul Aktar is an efficient officer and played a key role in apprehending militants," the minister told reporters in Chittagong.

Investigators in Chittagong said that they intercepted footage of closed-circuit cameras installed at nearby buildings showing three assailants carrying out the operation in under 50 seconds.

“The attackers first stabbed her [Mitu] and then shot her in the head from point blank to make sure she was dead before they fled,” disclosed Ghosh.

The killers had taken up position with the motorbike quite some time earlier and one of them was seen wearing a helmet, he said.

Unknown criminals stabbed Christian man Shunil Gomez, 60, to death at his grocery shop near his home in Boraigram in the northern district of Natore, police officer Shafiqul Islam said.

The assailants attacked Gomez in the neck with sharp weapons and quickly fled, investigator Doyal Kumar told local media at the scene.

The scene is about 140 kilometres north of Dhaka.

Kumar said the style of the attacks was similar to those that were carried out recently by suspected Islamists on bloggers, writers, a university teacher and Hindu men.

Islamic State, an extremist group controlling vast parts of Iraq and Syria, claimed responsibility for murdering Gomez.

US-based SITE-Intelligence group put on its website hours after the killing that the IS news agency Amaq reported IS fighters in Bangladesh killed a Christian in Bonpara village.

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