Child's father and mother were also injured in attack while her eight-year-old sister escaped unhurt
The girl's father and mother were also injured in the attack in the village of Saint-Herbot in Brittany while her eight-year-old sister escaped unhurt.
The Thornon family, who have lived in the village for about five years, were enjoying a barbecue in their garden and the girls were playing on swings at about 10pm on Saturday when a man appeared with a gun and fired several shots.
Solenne Thornon, 11, was killed immediately while her father, 52-year-old Adrien, was shot in the head. He was taken to hospital where his condition was described as critical.
The girls' mother, Rachel, 49, reported to be a local home help, was shot in the back and the head but was reportedly out of danger, the public prosecutor Carine Halley said. The youngest girl, Céleste, aged eight, escaped to a neighbouring property and raised the alarm.
Halley said a 71-year-old Dutch man who lived in the same village had been taken into custody in connection with the shootings. His wife was also being questioned.
French media reported there had been a conflict between the British family and their neighbours over a plot of land.
An investigation for the murder of a minor and attempted murder has been opened.
Halley said initial evidence suggested the suspect "emerged suddenly with a gun and fired several times in the direction of the victims".
"The motives for this tragedy are not yet known. It appears there had been a conflict between the two neighbours for several years over a piece of land adjoining the two properties," Halley told French journalists.
The killing came just days after a three-year-old British girl was among four children and two adults stabbed in a park in Annecy in eastern France.
Comment: Annecy, France: 4 children of nursery age stabbed in playground by Swedish-approved Syrian 'refugee'
Marguerite Bleuzen, the mayor of Plonévez-du-Faou, said the attack appeared to be connected to a "dispute between neighbours" over land and noise that dated back three years.
"We knew the family well. There is a village fete every year and they always came," a visibly shocked Bleuzen said.
"It's completely incomprehensible to have shot a child. This happened without any warning," she added. "Nobody can understand how this could have happened."
A neighbour told journalists: "We heard noises and thought they - either the children or the parents - were playing with fireworks. Then the youngest girl came running to the neighbours shouting: 'My sister is dead, my sister is dead.'
"We went and saw the girl was dead and her father and mother were injured."
Another village resident said: "The neighbours had complained about noise but they [the family] didn't make any more noise than anyone else."
Comment: Whilst these seemingly random knife attacks have been on the increase in recent years, this incident comes on the heels of a spate of attacks in recent weeks: