Three people got into a verbal dispute near the St Albans shop on Sunday morning before a shot was fired into the back windscreen of a blue Holden Commodore.
A man and woman inside the car fled towards Sunshine Hospital on Furlong Rd and crashed into a responding marked police vehicle.
They ran away on foot in different directions, with the woman arrested inside the hospital and the man a block away.
The woman suffered an injury to her foot but police are unsure whether it occurred in the shooting or in her efforts to evade officers.
Acting Senior Sergeant Ben McGibbon said the hospital was locked down while police searched a nearby construction site and surrounding areas. It resumed normal operations after about two hours.
The pair have been taken to a police station as detectives look to uncover what triggered the initial altercation.
"Not many things like this occur at this time of the morning on Sunday," Mr McGibbon told reporters.
"But for this to have occurred something has sparked it, and there's either been an altercation or disagreement or something has happened."
The third person remains on the run and police believe they may be carrying a small handgun.
"The investigation is in its infancy," Mr McGibbon said.
"We're still hoping for some other witnesses to come forward to tell us an independent version of what's actually occurred."
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