The city, which has frequently been on high security alert since the September 11, 2001 Al-Qaeda attacks on the Twin Towers, is America's financial and entertainment capital, with a population of 8.5 million.
"The vehicle struck multiple people on the path. There are several fatalities and numerous people injured," police tweeted.
"The vehicle continued south striking another vehicle. The suspect exited the vehicle displaying imitation firearms & was shot by NYPD."
Emergency services said the incident happened at 3:06 pm (1906 GMT) at Chambers and West Streets in the upmarket TriBeCa neighborhood, near Stuyvesant High School.
Police officers and ambulances swamped the area, in front of a park and next to a school as sirens wailed continuously and helicopters roared through the sky.
Worried parents gathered outside a public elementary school that had closed, waiting to see if they could collect children who stayed after the end of the classes for extracurricular lessons and activities.
"I didn't see the actual shooting, but I got there about 30 seconds after," said witness John Williams, 22, who was en route to the park. "There was a smell of gunshots.
"There was a man lying on the ground. It looked as he'd been shot," Williams added.
New York State Governor Andrew Cuomo and Mayor Bill de Blasio were briefed on the incident at the scene.
President Donald Trump has been briefed on the incident, the White House said.
"Our Thoughts and prayers are with all those affected," spokeswoman Sarah Huckabee Sanders said.
Retailer Home Depot confirmed that the truck was a rental. "We'll be cooperating with authorities in their investigation," a spokesperson told AFP in an email.
It was the first mass casualty incident in New York since a US Navy veteran plowed a car into pedestrians in Times Square, killing an 18-year-old woman from Michigan and injuring 22 other people on May 18.
Police patrols in Times Square, one of the world's busiest districts, had already been beefed up in recent years, following a 2010 car bombing attempt and in the wake of attacks in Europe.
A witness who gave his name only as Frank told local television network NY1 that he saw a man running around an intersection, heard five to six gunshots and saw "about 100 cops" flood into the street.
"I saw he had something in his hand, but I couldn't tell what it was. But they said that it was a gun, so I was like: 'Oh my God,'" the witness told NY1.
"When the cops shot him, everybody started running away and it got a little bit crazy right there. So when I tried to look again, the guy was already down."
The last confirmed terror-related incident in New York was a pipe bomb explosion in September 2016 in Chelsea, lightly wounding 31 people. An American of Afghan descent, Ahmad Khan Rahimi, was convicted of terrorism earlier this month.
The most serious security alert in the city since the 2001 attacks saw Faisal Shahzad, a Pakistani immigrant, plant a car bomb in Times Square on May 1, 2010.
His explosive device failed to detonate and he was arrested shortly after boarding a flight to the Middle East. He pleaded guilty and said he was aiming to avenge deaths from US missiles fired from drones operating over Pakistan.
He was sentenced to life behind bars.