The incident happened at about 2 p.m. Sunday at the store in Senatobia, Mississippi on U.S. 51.
Police said officers and the Tate County Sheriff's Office responded to a report of shoplifting at the local Walmart.
Mississippi Bureau of Investigation said when police arrived they found two people and and a child running out of the store into a car.
When police tried to stop the car, the driver allegedly tried to strike one of the police officers.
MSI said that's when an officer discharged their weapon and the driver fled.
The suspects then arrived to the hospital where the child died, another person had critical injuries.
No other information was available.