Derbyshire Police said a black Suzuki Swift was driven into pedestrians in Friar Gate at about 21:30 GMT on Saturday.
Seven people were seriously hurt but their injuries are not life-threatening.
Police described the suspect as a Derby man who is originally from India but has lived in the UK for a number of years. He is being held on suspicion of attempted murder, causing serious injury through dangerous driving, inflicting grievous bodily harm with intent and dangerous driving.
Derbyshire detectives are leading the investigation but counter-terrorism officers are assisting.
The force said this was “common practice for an incident of this nature” and officers were “keeping an open mind about the potential motives”.
The incident is not currently being treated as terrorism.
Ch Supt Emma Aldred said it had sent “shockwaves” through the community.
“Thanks to the fast actions of officers, working on eyewitness information from the public, we were able to quickly arrest the suspect in the city just a few minutes after the incident took place,” she said.
Aldred said it was a complex investigation and police would provide updates as and when they could.
The force also confirmed that “contrary to online speculation, no-one died in the incident”.
It added: “Although we know this will be alarming, we would like to reassure people that we do not believe there is an ongoing risk to the public.”
Education Secretary Bridget Phillipson told the BBC the home secretary was being kept updated on developments.
Mark Lucas, who lives near Friar Gate, saw the immediate aftermath and said there were people lying injured on the road.
The 47-year-old said: “I ended up coming down with my son and I realised it was a big incident.
“People were over the floor and there were a lot of police and ambulances.
“No-one really knew what actually happened. We didn’t know the car had run people over.”
Lucas described seeing paramedics trying to help two people who appeared to be “seriously hurt” and “weren’t moving”.
“I was with my son so I quickly sort of got him out the way,” he added.
Friar Gate has been closed from the junction with Curzon Street/Cheapside up to Ford Street, and people have been told to avoid the area completely.
The scene on Sunday was eerily quiet, with lots of police tape cordoning off the area and officers standing guard.
But it would have been a very different picture on Saturday night as the area is filled with restaurants and bars and is very popular with students.
Derby is a city but in many ways feels like a big town due to its community spirit.
There was a real sense of shock from people walking past the scene earlier, a feeling that this sort of thing does not happen around here.
Police said that cars parked in Friar Gate would not be able to be moved until the cordon had been lifted and the road reopened.
Inside the cordon, debris, including shoes and clothing, litters the road and forensic officers in white suits are walking along Friar Gate taking pictures.
Catherine Atkinson, Labour MP for Derby North, arrived at the scene and told the BBC: “Like many I remain really shocked. You don’t expect this to happen in our city.
“I think that everyone is utterly stunned by what has happened,” she said.
“Obviously the focus is on making sure that those who have been injured are looked after, but I’m hugely grateful for the swift action that we’ve seen from the emergency services.
“You see these incidents around the world but you never expect them to happen in your city, your home.”
She said she would be raising the subject of cars being driven at people in Parliament.
“I am concerned and I will be raising this with government,” she said.
“We need to see if guidelines will be reviewed because we are seeing an increased amount of incidents where cars are being used as weapons.”
Prayers are being held at Derby Cathedral and in churches across the city for the victims.
Rt Rev Malcolm Macnaughton, Bishop of Repton, said the incident had “shaken us all”.
“We like to think of our city as a safe place for everyone,” he said.
“Our prayers… are with those injured and with all those who witnessed what happened.
“We pray also for those receiving treatment in hospital. May God bring peace and healing to our city.”
O’Dwyers Bar, in Friar Gate, said on social media it would remain closed on Sunday.
It said: “We are deeply saddened by the horrific incident that took place on Friar Gate last night.
“Our thoughts are with all those injured, their families, and everyone impacted by what has happened.
“No-one should ever have to experience something like this on a night out in our city.
“Derby is a community, and today we stand together.”
Nadine Peatfield, leader of Derby City Council, said she was “horrified” by the incident.
“My thoughts, and those of the entire city, are with those injured, their families, and everyone who witnessed this distressing event,” she said in a statement.
“We are working closely with Derbyshire Police as they investigate the circumstances.
“Please continue to avoid the affected area to allow officers to carry out their vital work.”
Education Secretary Bridget Phillipson told the BBC’s Sunday With Laura Kuenssberg: “My thoughts are with anyone that’s been affected by this, those who’ve been injured or hurt, and their family members at this really difficult time.
“The police are actively investigating. We don’t believe there is wider cause for the public to be alarmed.
“The police have made an arrest, and the home secretary is being kept updated on this.”
Police forces release the ethnicity and nationality of suspects in high-profile cases.
The National Police Chiefs’ Council said the aim was to reduce the risk to public safety where there are high levels of misinformation about an incident.
