A six-year-old is in hospital with life-threatening injuries after apparently being shot in the head by her father, who then turned the gun on himself.
The body of a man - believed to be the girl's father - has been found in a car close to the scene of the shooting at an address in Spring Hill, in Northiam, near Rye, East Sussex.
Police say the death is not being treated as suspicious.
The girl has been taken to King's College Hospital where police say she is fighting for her life.
Chief Superintendent Neil Honnor said: "We are treating this as a tragic domestic incident and do not believe that anyone else is involved.
"At this stage we are not in a position to confirm the relationships of all those involved but we can say that the girl and the man knew each other. The man does not live at the address. We are treating this as an isolated incident and there is no evidence that any other members of the public are at risk."
Mr Honnor said that "we probably need to prepare ourselves for bad news" in relation to the young girl's condition.
He told BBC Radio Sussex: "She's still alive, but gravely, gravely ill and our hopes are with her but in many ways we probably need to prepare ourselves for bad news."
An earlier statement from the Sussex force said there was "no suggestion that the public are in danger" and that they were not looking for anyone else.
Armed officers were sent to the scene after the shooting, with scores of police cars surrounding an area cordoned off to the public.
Villager Anthony Burr, 37, lives near the scene of the shooting.
"I was in the living room and heard the sound of a firearm," he told Sky News.
"The sound of police sirens followed some minutes later. He said a neighbour had told him the girl had been shot in the head. Obviously there was panic across Northiam, we all went into lockdown, shut doors, got the children in," he said.