Police said the incident occurred at 2130 GMT on Saturday.
All seven people suffered serious but not life-threatening injuries and were treated at the scene before being taken to hospital.
"We can confirm that seven people were injured when a car was driven into pedestrians in a city centre street," police said in a statement.
"Seven people were treated at the scene by ambulance crews and taken to Royal Derby Hospital and Queen's Medical Centre in Nottingham."
Police said officers stopped the vehicle believed to be involved shortly after the incident and arrested the driver, a man in his 30s, who remains in custody.
The man, originally from India, was arrested on suspicion of attempted murder, causing serious injury by dangerous driving, inflicting grievous bodily harm with intent and dangerous driving.
Police said their investigation into the circumstances surrounding the incident was ongoing.
"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," the police said.
Derby is in the East Midlands region, 180km north of London.
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