A 29-year-old man, believed to be the driver, with German-Armenian dual citizenship was detained, German police told Sky News.
Iris Spranger, the city's interior affairs minister, suggested the motorist may have had mental health issues.
She said: "The latest evidence suggests this is a case of a mentally ill person running amok."
Berlin police spokesman Martin Dams said earlier it was unclear whether the incident was an accident or the driver had intentionally ploughed into the crowd.
The man had been driving a small Renault car, Bild reported.
According to witnesses, he initially ran away and was stopped and handed over to police, it added.
The children are from the small town of Bad Arolsen in the central German state of Hesse, and are reportedly in the 10th grade, meaning they are aged 16 and 17. Their female teacher died at the scene.
The German chancellor Olaf Scholz said the "horrific act" on a busy street had deeply affected him and he hoped those caught up in the crash make a quick recovery.
He tweeted: "The trip of a school class from Hesse to Berlin ends in a nightmare. We think of the relatives of the dead and the injured, including many children. I wish all of them a speedy recovery."
The fire service earlier said a dozen people were injured and six were in a life-threatening condition.
An apparent confession letter was found in the car. Germany's Bild newspaper cited an investigator as saying: "(This was) by no means an accident - someone on the rampage, an ice-cold killer."
But officials denied there was a confession letter, instead claiming posters about Turkey were apparently discovered in the vehicle. Turkey has strained relations with Armenia.
Footage of the scene showed the silver car had crashed into the window display of a department store.
Blankets covered what appeared to be a body in a cordoned-off area guarded by police.
Large numbers of paramedics and ambulances were at the scene.
Reporting from the scene, Sky News' Europe correspondent Siobhan Robbins said the car went down the street and then hit a group of people, went back on to the road, mounted the kerb again before crashing into a shop.
It took place at about 10.30am on Rankestrasse, near the main shopping district's Breitscheidplatz, the scene of the 2016 Christmas market attack in which 11 people were killed.
This is also near the war-ravaged Kaiser Wilhelm Memorial Church, one of the city's best-known landmarks.
Actor John Barrowman, who was at the scene with his husband Scott Gill, told Sky News he saw "a dead person in the middle of the road, covered up".
"The person came on the pavement by the Levi's store and the bank, then drove onto the street, hit the person and killed them in the street, then came back onto the kerb, went down the kerb, through the cafe area and then into the storefront," he said.
"How horrible that this morning these people got up to just go through a regular day, they were going through their daily business. I mean we were shopping, but if you think if we'd have come out of that store one minute earlier..."
He said: "We are no experts but this looks intentional - by the way the car went on the pavement."
"There were chairs and tables smashed on the pavement - just a trail of destruction," said Mr Gill, a British architect.
"You could see the car had zig-zagged on and off the pavement on the way down."
"For the first 10-20 minutes it was just sheer chaos," he added. "We walked right through it without really realising what was happening."
"We are quite shocked," added Barrowman, fighting back tears.
"To see people hurt, people who were just innocent and the senseless destruction by whoever has caused this.
"Whoever has done this is a horrible, horrible person... it is just a vicious thing to do."
"We think we have witnessed a terrorist attack here in Berlin," he tweeted earlier.
"We've seen a car that came down the road and ended up in a storefront covering three city blocks. It's pretty horrific."
"There are police behind us, there are other police cars coming. There's helicopters coming in. This is bad. This is serious," he continued in video messages posted on social media.
"There are hundreds of emergency services that are in the area, blocking city block after city block, off. The cordon keeps expanding."