A Polish man who was supposed to have been expelled from France has admitted to kidnapping and killing a little girl in the northern town of Calais after she shot a water pistol at him.

Police were questioning the 38-year-old unidentified suspect on Thursday, a day after the killing.

Prosecutor Jean-Pierre Valensi told reporters the body of 9-year-old Chloe showed she was strangled and sexually assaulted. Valensi said the suspect admitted to the killing.

Authorities are trying to establish how a man convicted in 2009 and outlawed from France may never have left.

Justice Minister Christine Taubira said he was handed over to Polish authorities last year. She demanded a "complete, precise, rapid" investigation.

"What we owe the family," said Prime Minister Manuel Valls, "is the entire truth."