Los Angeles police have arrested a suspect who earlier barricaded himself in a shop with about 40 people inside.

"We have successfully taken the suspect into custody without incident," the LAPD tweeted.

People were earlier seen running from the Trader Joe's supermarket. There was one fatality, police said.

The suspect earlier fired shots after police pursued his car. He crashed the car, running inside the shop. The chase began after he shot two women.

The suspect - who is believed to be 28 years old - gave himself up after about three hours of crisis negotiations with police officers, the LAPD said.

He asked that he put handcuffs on himself before leaving the shop. He had an arm injury.

A young woman inside or near Trader Joe's was shot and killed by the suspect, police said. She has not been identified.

Earlier, another woman was taken from the crashed car and taken to a local hospital.

She is believed to have been injured and forced into the vehicle after the gunman shot his grandmother.

The grandmother was shot a number of times, and was in a critical condition, police said.

US President Donald Trump tweeted that he was watching the situation "very closely".