The girl, named Solenne Thornton, was reportedly killed while playing in the garden of her home in Saint-Herbot, a village near the city of Quimper in Brittany, at about 10pm local time on Saturday.

According to local television network France 3, her father, Adrian Thornton, is in a critical condition, while her mother has non-life-threatening injuries. Both parents are reportedly in their 40s.

The victim's mother and sister were named as Rachel and Celeste Thornton.

The girl's sister - who is eight years old - managed to escape unhurt and raised the alarm, but is in a state of shock.

It is believed the family had lived at the property for four or five years - and both girls were born in France.

The suspect, a 71-year-old Dutch national, has been arrested on suspicion of killing the girl, according to local prosecutors. His wife is also reportedly being held.

Local media has reported the family was having a barbecue at the time of the attack.

Yannick, a local resident, told the daily regional paper Le Telegramme de Brest: "We heard the bangs and we thought it was children playing with bangers. Then the kid ran into a neighbour's house saying: 'My sister is dead, my sister is dead'."

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According to local prosecutors, initial evidence suggests the pensioner "suddenly emerged armed with a gun and fired several times towards the victims".

He is reported to have then shut himself in his house with his wife.

The Intervention Group of the National Gendarmerie tactical unit was called to the scene and a negotiator persuaded the neighbour and his wife to give themselves up before they were arrested, Carine Halley, the state prosecutor in Quimper, reportedly said.

She added that the attacker's motive is not yet known but it would appear the two sets of neighbours had been engaged in a long-running dispute over a plot of land adjoining their properties for several years.

One neighbour said the dispute with the Dutch pensioner began when they "cut down all the trees that bothered them".

A neighbour told La Telegramme he had seen the whole family under the oak tree at the bottom of their garden earlier on Sunday evening.

"The two little girls were playing... the parents were sitting on their usual bench and having a barbecue," he said.

Plonévez-du-Faou's mayor, Marguerite Bleuzen, said: "We knew the family well. There is a village fête every year and they always came.

"They are not a family who caused problems. Perhaps there was a dispute with the neighbour, I don't know about that. But it is incomprehensible to have shot a child. No one can understand how that could have happened."

A spokesperson for the Foreign, Commonwealth and Development Office said: "We are providing consular assistance to a British family following a shooting in France and are in contact with the local authorities."