Jean-Marc Duprat, deputy mayor for the town of Laragne-Monteglin in the Hautes-Alpes region, said the mother and her girls, aged eight, 12 and 14, were on holiday at a nearby resort when a man from a neighbouring apartment attacked them yesterday morning.
Mr Duprat initially said the suspect was upset by what the girls were wearing, but later said that did not appear to be the case and the motive was not known.
The youngest victim is in a critical condition in hospital after suffering a punctured lung.
The injuries suffered by the other three victims are serious but not life-threatening, hospital sources said.
Laragne-Monteglin is 110 miles north-west of Nice, where a Tunisian man killed 84 people last week by driving through a crowd on Bastille Day.
The suspect was named by police as Moroccan-born Muslim Mohamed Boufarkouch, 37, who lives in Yvelines, France. He was holidaying with his own wife and two children.
He fled after the attack, leaving behind his family and his wife who ‘was in tears’, before police later caught him.
It is understood the man was well known to police.
Despite Mr Duprat’s denial that clothing was a motive, various media outlets reported that the woman and her daughters were wearing shorts and T-shirts at the time of the stabbing. Reports said Boufarkouch attacked the women because they were scantily dressed’.
A French prosecutor, however, told the AFP news agency “the motive of the attack is very blurred.”
French television channel TF1 reported the victims may have acted out of religious motives.”
French media also said the man was not on the so-called ‘Fiche S’ list, associated with radicalised individuals.
TF1 also reported that the stabbed mother had helped the attacker when he became ill a day earlier.
It is understood he first stabbed the girls while they ate breakfast on the terrace of their chalet and then went inside and stabbed their mother in the chest.
Media outlets said the attacker was on holiday with his pregnant wife and two children. The assailant’s wife is “devastated”, the mayor said. Local newspapers reported that the youngest girl was helicoptered to hospital in Grenoble as an emergency case.