Kamilah Peniston from the Gorton area of Manchester was taken to Manchester Royal Infirmary at 19:30 BST and then later transferred to Pendlebury children's hospital, where she died this morning.
Greater Manchester Police recovered a handgun converted to fire live ammunition at the same address where she died.
Detective Superintendent Ian Foster, who is leading the investigation, said the events leading to her death were "tragic".
"Our deepest sympathies are with Kamilah's family at this sad time and we will be doing everything possible to find out what the circumstances were that led to her death," he added.
"One thing I would like to stress is that this tragic incident shows the devastating impact that guns can have on our community.
"I would urge anyone who has any information about people who are selling guns, converting guns or who have them in their homes, to contact us. We need to take these weapons off the streets before another innocent life is needlessly taken."
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