It happened around 3:45 p.m. Tuesday on the 1600 block of Telegraph Road in Stanton, New Castle County.
Police say a 44-year-old woman got out of her car to exchange information with another driver after a minor collision.
There was allegedly an argument between the two, and the other driver took off, hitting the woman, according to police.
Police then say the pedestrian was taken to an area hospital for non-life-threatening injuries.
Authorities are now searching for the driver, who could be driving a dark-colored newer model SUV, with some reports saying it may be a Honda or Hyundai.
Anyone with information on the driver is asked to contact Detective Belk at Corey.Belk@newcastlde.gov or call (302) 395-8077.