Solaine Thornton and her sister Celeste, who is eight, were in the garden at their home in the village of Saint-Herbot, near Quimper, Brittany, when an elderly neighbour allegedly opened fire at them with a shotgun through a hedge, according to reports.
The suspect, a 71-year-old Dutchman, is also reported to have been in a long-running row with the family over a plot of land beside their properties.
Carine Halley, the state prosecutor in Quimper, said Solaine was dead when police arrived.
The girl and her parents, Adrian and Rachel Thornton, were reportedly named by mayor of Plonevez-du-Faou Marguerite Bleuzen.
They are both reported to have been shot in the head.
Mr Thornton is understood to be in a critical condition.
An official at Plonévez-du-Faoum - the local town hall for the family’s village - confirmed that Adrian and Rachael were born in Oldham and that Solaine and Celeste were born in France.
The UK Foreign Office said it was aware of the shooting and has offered consulate assistance.
