Lt-Col Arnaud Beltrame, 44, was shot and stabbed after he traded places with one of the captives following a shooting spree in southern France.
Flags are being flown at half-mast at gendarmerie bases across France.
His brother Cedric said Col Arnaud “didn’t have a chance”, adding that his actions were “beyond the call of duty”.
“He gave his life for strangers. He must have known that he didn’t really have a chance. If that doesn’t make him a hero, I don’t know what would,” Col Arnaud’s brother Cedric told a French radio station on Saturday.
Speaking to the BBC, Col Arnaud’s cousin Florence Nicolic described the officer as a person who was “so good at his job”.
“Even though we were surprised and shocked when we heard what happened we were not surprised in the sense that that’s the thing he would do without hesitation,” Ms Nicolic said.
French President Emmanuel Macron also paid tribute to the officer, saying that Col Arnaud “fell as a hero” after showing “exceptional courage and selflessness”, adding that he deserved “the respect and admiration of the whole nation”.
UK PM Theresa May said the “sacrifice and courage” of the police officer would not be forgotten.
His actions helped bring an end to the siege that left three people dead.
The radical Islamist gunman, 25-year-old Redouane Lakdim, was eventually shot and killed by police.
Sixteen people were injured, two seriously, in what Mr Macron called an act of “Islamist terrorism”.
Lakdim was said to have demanded the release of Salah Abdeslam, the most important surviving suspect in the 13 November 2015 attacks in Paris, which killed 130 people.
Prosecutors are reportedly questioning two people in connection with the attacks, one of whom is thought to be the gunman’s partner while the other is believed to be a friend.
The violence began on Friday morning in Carcassonne, where Lakdim hijacked a car. He killed a passenger – whose body was later found hidden in a bush – and injured the driver.
He then shot at a group of policemen who were out jogging, wounding one of them.
Lakdim is then believed to have driven a short distance to the small town of Trèbes, where he stormed into the Super-U supermarket, shouting, “I am a soldier of Daesh [Islamic State]!”
He killed two people – a customer and a store worker – before seizing others as hostages.
Collomb told reporters on Friday that police officers had managed to get some people out of the supermarket but the gunman had held one woman back as a human shield.
It was at this point, he said, that Col Beltrame had volunteered to swap himself for her.
As he did so, he left his mobile phone on a table with an open line so that police outside could monitor the situation.
When police heard gunshots, a tactical team stormed the supermarket. The gunman was killed and Col Beltrame was mortally wounded. (Courtesy BBC)