Faisal Arefin Dipan was murdered at his office in central Dhaka, Imran Sarker, who heads a secular bloggers group, told AFP.
"He was hacked to death. I saw him dead," he said.
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Police confirmed that Dipan was seriously injured in the attack and was brought to Dhaka Medical College Hospital.
"They still have not declared whether he is dead," said police inspector Mozammel Haq.
Dipan is the son of a noted Bangladesh writer Abul Kashem Fazlul Haq and owns Jagritee Publishers.
Earlier, two secular writers and the publisher of an atheist blogger killed earlier this year by suspected extremists were attacked on Saturday in Dhaka, police and activists said.
The three are the latest victims in a series of attacks against the country's secular activists, which police have blamed on the banned militant group Ansarullah Bangla Team.
"Some unknown miscreants attacked Ahmedur Rashid Tutul, the publisher of a slain atheist writer, secular blogger Ranadipam Basu and Tareq Rahim at the office of Shuddhaswar publications in the capital," local police chief Jamaluddin Mir told AFP.
Fourth secular blogger hacked to death in Bangladesh
Activists described the third victim, Rahim, as a young secular poet.
"The condition of at least one person is critical," he said, adding that the attackers padlocked the office door from outside as they left.
Police said officers broke the lock and found the three injured men in a pool of blood.
"They have been rushed to a hospital," Mir said.
One of the victims Basu posted a short Facebook status immediately after the attack: "They hacked us, me Tutul and Tareq" .
"Doctors told us that the conditions of two, including the publisher, were critical. They were attacked in their heads and chests," Imran H. Sarker, the head of a secular bloggers' group told AFP, adding two of them appeared to also have bullet wounds.
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Sarker said that Tutul was the owner of Suddhaswar, which published books by Avijit Roy and several other young secular writers.
Roy, a US national of Bangladeshi origin, was hacked to death by suspected militants outside a book fair in Dhaka in February this year.
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