A woman was fatally wounded when a gunman being chased by police in Los Angeles fled into a supermarket and took hostages, authorities have said.

Los Angeles Mayor Eric Garcetti said the women was killed inside the Trader Joe’s store in the Silver Lake area.

A man was taken into custody following the incident.

Police said the suspect shot his grandmother and girlfriend on Saturday before leading police on a car chase that ended when he crashed into a pole and ran inside the supermarket as bullets from officers shattered the front doors.

About three hours after he took hostages in the store, the suspect surrendered.

Officers with riot gear, armed with rifles, had stood along the side of the Trader Joe’s and used mirrors to try to look inside as hostages periodically came out of the front door with their hands raised.

The suspect walked out with a cluster of four hostages and appeared to be already handcuffed as the group emerged through the front door.

Police immediately surrounded the suspect, searched him and then brought him to a waiting ambulance. The man appeared to have blood on his left arm.

Investigators believe the suspect, whose name has not been released, had shot his grandmother and girlfriend at about 1.30pm in South Los Angeles and then fled in a 2015 Toyota Camry, Sergeant Barry Montgomery, a Los Angeles police spokesman, said.

Officers spotted the suspect’s car near Hollywood and tried to pull him over, but the man refused to stop and led officers on a pursuit, he said. During the chase, the suspect shot “multiple rounds” at officers, though no officers were struck by the gunfire, he added.

The suspect eventually crashed his car outside the supermarket and then ran into the store.

Don Kohles, 91, was walking into the supermarket when he saw a car being chased by police crash into a pole just outside. Police fired at the driver, shattering the store’s glass doors, while those inside the store took cover and laid on the floor as the suspect ran inside, he said.

He could hear others around him sobbing as the man ran towards the back of the store and yelled at people, but Mr Kohles said he did not hear any more gunshots. After about 30 minutes, police came inside and rushed some of the customers out, he said.

Police officers use a mirror to see inside a Trader Joe’s store
Police officers use a mirror to see inside a Trader Joe’s store (Christian Dunlop via AP)

People frantically tried to flee from the store and some were seen climbing through windows while others darted through the back door.

Christian Dunlop, an estate agent and actor who lives nearby, said he was watching from the corner when he saw four people run out the front of the store. An employee was dragging an injured woman by the hands out of the front door, he said.

One woman who was injured was taken to the hospital in a stable condition, according to David Ortiz, a fire department spokesman, although it was unclear how she was injured. Officials said they had 18 ambulances and 100 firefighters at the scene.