A search was underway for a suspect who fled on foot after the blast late on Monday, Prosecutor Stéphane Thibault said, adding that the motive was unclear.
Media reports identified Ukrainian construction tycoon Vadym Yermolaiev as being among the injured. He has said he renounced his Ukrainian citizenship nearly a decade ago, and he was targeted by Ukrainian sanctions in 2023 for ties to Russia. A woman and a child were also hurt.
"It appears that the family was specifically targeted," said Christophe Mirmand, the minister of state for Monaco.
The suspect "had walked around the area several times while waiting for the victims", according to surveillance footage, Mirmand said.
It was not clear why the family was targeted or by whom.
Russia has a long history of targeting its enemies abroad, and Western intelligence officials have recently said that a campaign of targeted killings has ramped up since President Vladimir Putin's 2022 invasion of Ukraine.
Ukraine is also believed to have carried out attacks and targeted killings of Russian figures in the course of the war, although those attacks have largely been confined to Ukrainian or Russian territory.
The Ukrainian Foreign Ministry said its embassy in France, which is also responsible for Monaco, was in touch with local authorities. Ukrainian diplomats were at the scene providing assistance, it said.
The attack shocked the country on the Mediterranean coast, one of the world's smallest sovereign states known for its high concentration of wealthy residents.
Monaco's Prince Albert II described it as "an odious act" and said all public services were mobilised to ensure security.
Monaco police opened an attempted murder investigation, but they did not describe it as a terrorism investigation, Thibault told reporters.
The family members are "regular" residents of Monaco, and authorities did not yet know whether they had been threatened in the past, Mirmand said.
Silvano Ippolito, who lives across from the building where the explosion happened, described hearing it and seeing a little boy on the ground being attended to by other people. He then called his wife, a doctor, who provided first aid to the badly wounded woman.
"She intervened very quickly, before the emergency services arrived, to apply tourniquets and perform mouth-to-mouth resuscitation, as the woman was losing consciousness," Ippolito said.
A man came out of the building covered in blood and staggering. As he tried to go down the stairs, the staircase collapsed, and he fell on Ippolito's wife and a firefighter, he said.
The woman's injuries were "a real disaster," Ippolito said.
"I'm telling you, she had no foot."
The woman was in life-threatening condition.
Her partner and the 13-year-old child suffered less severe injuries but were still hospitalised Tuesday, he added.
The suspect got away via steps to a small street to the neighbouring French town of Beausoleil, according to surveillance footage.
In a picture captured by surveillance cameras and published by French media, the suspect could be seen in a street wearing a black jacket, light-coloured pants, white shoes and a black bucket hat that partly concealed his face.
The three people were "apparently returning home peacefully", Mirmand said.
"They were caught in the explosion as they crossed the threshold of their apartment building."
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