The suspected perpetrator has been “neutralised”, they said. Palestinian media showed photos of the suspect lying dead near the scene.
One of those seriously injured is a 46-year-old woman, the ambulance service said.
“The suspect was driving a vehicle travelling from south to north, rammed into pedestrians standing in the shopping centre and proceeded to get out of the vehicle to stab civilians with a sharp object,” the police said.
Militant group Hamas claimed responsibility for the attack, which it said is “the first response to the occupation's crimes against our people in Jenin Camp”.
It identified the assailant as Abdelwahab Khalayleh, 20, from the town of Al Samu near Hebron.
An Israeli operation in Jenin, in its second day, has killed 10 people and wounded 50 others, 20 critically.
UN aid agencies voiced alarm at the scale of the Israeli military operation in the West Bank town of Jenin, saying there were restrictions on medical access.
UN Secretary General Antonio Guterres also expressed deep concern about the developments
In a statement from his spokesman, Mr Guterres said all military operations must be conducted with full respect for international humanitarian law.
At least 3,000 people were forced to flee their homes in Jenin refugee camp as a result of the raid, which reduced much of the densely populated area to rubble.
Mayor of Jenin city Nidal Obeidi said: “The humanitarian situation in Jenin is catastrophic.”
“What’s happening is like an earthquake. It reminds us of the days of Nakba."
“We left with nothing but the clothes on our back,” Duha, 14, told The National, saying Israeli forces raided her home while her family were still there.
“We are used to Israeli attacks in Jenin camp, but nothing like this. I never thought I'd leave my home this way.”
The Jenin Brigades militant group called on the public to “declare their joy in celebration of this divine revenge” in Tel Aviv in response to the Israeli raid.
Mr Khalayleh was shot dead by an armed Israeli civilian, according to reports from Israel's Kan public broadcaster and the official Wafa Palestinian news agency.
At least 188 Palestinians, 25 Israelis, one Ukrainian and one Italian have been killed this year, according to an AFP tally compiled from official sources from both sides.
They include, on the Palestinian side, combatants and civilians, and on the Israeli side mostly civilians and three members of the Arab minority.