George Floyd's 4-year-old niece Arianna Delane was shot in the torso while sleeping in her bed in Houston on New Year's Day.
The shooting is thought to have been a "targeted attack" and happened just before 3am local time on Saturday. Four adults and two children were in the apartment.
The family drove Arianna to hospital. She has had surgery and is believed to be in stable condition.
She suffered a punctured lung and liver and several broken ribs.
"My daughter jumped up and said she had been hit," the girl's father told local media. "I [saw] the blood, the bleeding, and I grabbed her."
According to Houston Police Chief Troy Finner, an internal affairs investigation into the department's handling of the shooting is under way.
"I am aware and have concerns regarding the delayed response time and have initiated an Internal Affairs investigation," Finner said.
"I ask the city continue to pray for this child's full recovery and assist in providing information that would lead to the arrest of the suspect or suspects responsible," he added.
Police have no suspect description or motive at this stage.
Arianna had attended demonstrations held in honour of Floyd, who died on May 25, 2020, after former police officer, now convicted murderer, Derek Chauvin kneeled on his neck for more than nine minutes.
Floyd's death sparked worldwide Black Lives Matter protests against racial injustice and police brutality.