Mr. Saipov, a slim, bearded man, was seen in videos running through traffic after the attack with a paintball gun in one hand and a pellet gun in the other. Six people died at the scene and two others died at a hospital, officials said. The authorities credited the officer who shot him with saving lives.
"He was Johnny-on-the-spot and he takes the guy down,” a city official said.

Coming five months after a car rammed into pedestrians in Times Square, killing one, Tuesday’s attack again highlighted the danger of a car attack on busy city streets. The Times Square incident was not a terrorist attack. But both incidents brought to mind the terrorist attack last year in Nice, France, in which a cargo truck killed scores of people celebrating Bastille Day.

The episodes also evoked calls from terrorist magazines, including in a recent edition of Rumiyah, a magazine used by ISIS, for attackers to mow down pedestrians with trucks, continue the attacks with a knife or a gun and claim credit by shouting or leaving leaflets.

Students in Halloween costumes streamed out of nearby schools after lockdowns were lifted and huddled with parents. Their faces, once painted for the holiday, were streaked with tears.
Emily, 12, a seventh-grader at I.S. 289 whose father asked that her last name not be printed, had been walking on her usual route home when other students turned and ran in the other direction.

"All the kids were screaming, ‘Run!’, ‘Gun!’ ‘Run inside,’” she said, still wearing cat ears. She said mothers pushing strollers and children in costumes ran in a herd back toward the school.
President Trump responded to the attack on Twitter: "In NYC, looks like another attack by a very sick and deranged person. Law enforcement is following this closely. NOT IN THE U.S.A.!”

Gov. Andrew M. Cuomo cautioned at a news conference, "There’s no evidence that suggests a wider plot or a wider scheme.” In the aftermath, city and state law enforcement agencies increased security at high-profile locations.

Terrorism analysts noted that on Monday a French pro-ISIS media unit, known as the Centre Mediatique An-Nur, put out a specific threat for Halloween, mentioning the date on a banner spread on the encrypted app Telegram and on ISIS-affiliated Twitter accounts.

In chat rooms with ISIS followers, supporters cheered the Tuesday’s attack. At the same time, ISIS members were trying to discern if the attacker was one of their supporters.

The Islamic State’s official media outlets made no mention of the violence in Manhattan. In the past, the terrorist group has generally not claimed attacks when the perpetrator is in custody, as was the case in the Manhattan truck attack.

Mr. Saipov wove a deadly path on a stretch usually bustling with commuters, runners and cyclists, drawn by the downtown offices nearby or the shimmering river.

He turned onto the bike path alongside the West Side Highway at Houston Street just after 3 p.m. and sped south, striking numerous pedestrians and cyclists, many of them in the back, the authorities said. People scattered and dove to the asphalt.

The truck, labeled with a sign saying, "Rent me starting at $19,” rammed into the bus near Chambers Street. The bus serves two schools in Lower Manhattan and transports students with special needs. Two adults and two children on the bus were injured, the authorities said.

Mr. Saipov jumped out of the truck before a uniformed officer assigned to the city’s First Precinct shot him, Police Commissioner James P. O’Neill said. The police said they were not looking for additional suspects.

Officials said the 11 people were taken to nearby hospitals with serious, but not life-threatening, injuries.