The incident took place at around 8.55pm on Thursday, March 17 on Grange Avenue in Normanton, Derby.
A young child was with his parents who were getting into their car, when he ran out into the road and was hit by another vehicle.
He was taken to hospital for treatment for injuries which were not found to be serious.
The vehicle, which is described as being a blue Vauxhall Zafira or similar, allegedly initially pulled over but then drove off onto Village Street.
Officers have carried out house-to-house enquiries and pursued other routes of investigation. They are keen to hear from anyone who may have dashcam footage or information which could help to identify the driver.
If you think you may be able to help, please contact Derbyshire Police using the non-emergency details below, quoting reference 22*155268:
Facebook – send a private message to /DerbyshireConstabulary
Twitter – direct message the force’s contact centre on @DerPolContact
Website – complete the online contact form
Phone – call 101
You can also anonymously contact Crimestoppers on 0800 555 111.