The incident occurred in the eastern Pakistani city of Chakwal on June 10.
The child, identified as Hania Ahmed, died on the spot, while her father, Adeel Ahmed, 39, and 10-year-old brother, Aafan Ahmed, sustained critical gunshot injuries and remain under treatment.
Adeel's wife, Sidra Khan, escaped unharmed.
According to local police official Sajjad Hussain, personnel from the Crime Control Department (CCD) mistook the family's vehicle for that of suspected criminals and opened fire, Arab News reported.
Police initially registered a case of murder and armed robbery against unidentified suspects. However, following public outrage and a preliminary investigation, authorities amended the case and arrested CCD constable Shujaat Mughal on murder charges.
The incident drew international attention, including from Australian Prime Minister Anthony Albanese.
Local police said two alleged dacoits were also killed during the encounter.
According to reports, the ordeal began when armed robbers hijacked the family's rental car at gunpoint on the night of June 10.
Police said the shootout erupted after the suspects opened fire at officers.
Recalling the incident, Adeel Ahmed said the gunmen intercepted their vehicle near the CCD office around 11:40 p.m. and demanded valuables, according to Arab News.
"My wife and I complied... My wife took off her jewelry and handed it over," Ahmed said in the First Information Report (FIR).
"While these individuals were robbing us of our jewelry at gunpoint, a shot was fired. Taking cover behind the car, the individuals started firing, and cross-firing also commenced from the front side," he said.
According to the FIR, Ahmed attempted to drive away to protect his children, but the vehicle was struck by a hail of bullets from behind, injuring him and both children.
Australian Prime Minister Anthony Albanese called for a transparent investigation into the circumstances surrounding the incident.
"These circumstances do need to be examined. They need to be examined in a transparent way, so that everyone can know — the family, most importantly, but others as well," Albanese told reporters in Canberra, according to the BBC.
"Australia expects there to be transparency and a proper investigation of these circumstances," he added.
Meanwhile, Abdullah Khan, principal of Hania's school in Perth, the Australian Islamic College, described the news of her death as "traumatic" for the community.
"Hania was very friendly, bubbly, and very social," Khan told the BBC.
"She had lots of friends and was very respectful to teachers. She was loved by everyone," he added.