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.
So-called Islamic State has said it was behind the van attack in Las Ramblas, saying in a statement carried by its Amaq news agency that it was carried out by “Islamic State soldiers”. The group gave no further evidence or details to back this claim.