The vehicle sped along the pedestrian area, mowing down people and sending others fleeing for cover.
Police are treating it as a terrorist attack.
The head of the Catalan region said two people had been arrested.
Spanish media are reporting another suspect died in a shootout. El Mundo newspaper said the shootout happened in Sant Just Lesvern, on the outskirts of the city.
Police have released a photo of a man named as Driss Oubakir, who is alleged to have rented the van used to drive into pedestrians.
