© Olafur Steinar Gestsson/APPolice have appealed for anyone who has seen, heard or filmed anything to contact them.
The number or condition of all the people struck by shots was not immediately available amid confusion over what had occurred at one of Scandinavia's biggest malls.
Copenhagen police chief inspector Soren Thomassen said authorities had no indication that there was more than one shooter. He said the suspect in custody was a 22-year-old Danish man, and that "terrorism" could not be ruled out as a motive.
"We know that there are several dead" and "several injured", Thomassen told reporters at a press conference. "We do not have information that others are involved. This is what we know now."
The capital's main hospital, Rigshospitalet, has called in extra staff, including surgeons and nurses, a spokesman said.
In a series of tweets earlier on Sunday, police said they had sent officers to a large mall in the capital named Field's, after receiving reports of a shooting.
Police were present in large numbers, they added, advising people inside the centre to stay put and await assistance.
"We have a massive presence at Field's and are working on getting an overview," police tweeted.
Images on social media that could not be immediately verified showed police running into a building and getting dressed in protective gear.
Other footage posted online showed people running through and out of the mall, and Denmark's TV2 broadcaster posted a photo of a man being put on a stretcher. Witnesses said people were crying and hiding in shops. Several fire vehicles were parked outside the mall.
Police have appealed for anyone who has seen, heard or filmed anything to contact them.
Copenhagen's mayor, Sophie H Andersen, tweeted: "Terrible reports of shooting in Fields. We do not yet know for sure how many were injured or dead, but it is very serious."
Laurits Hermansen told Danish broadcaster DR that he was in a clothing store at the shopping centre with his family when he heard "three or four bangs. Really loud bangs. It sounded like the shots were being fired just next to the store."
Other witnesses quoted by Danish media said they had seen more than 100 people rush towards the mall's exit as the first shots were heard.
"We could see that many people suddenly ran towards the exit and then we heard a bang. Then we ran out of Field's too," Thea Schmidt, who was in the mall at time of the attack, told broadcaster TV2.
Field's shopping centre, on the outskirts of the capital, has 135 shops, cafes and restaurants.
The mall is just across from a subway line that connects the city centre with the international airport. A major highway also runs adjacent to Field's.
A concert by British singer Harry Styles, which was due to take place at the Royal Arena less than a mile from the mall, was cancelled on Sunday evening despite earlier reports that it would go ahead.
On Snapchat, Styles wrote: "My team and I pray for everyone involved in the Copenhagen shopping mall shooting. I am shocked. Love H."
Shortly after the shooting, the royal palace said a reception with Crown Prince Frederik connected to the Tour de France cycling race had been cancelled. The first three stages of the race were held in Denmark this year, the palace said in a statement.
The reception was due to be held on the royal yacht that is moored in Sønderborg, the town where the third stage ended.
Comment: More footage has emerged on Twitter: