Donald Trump said he had ordered the Department of Homeland Security to step up its “extreme vetting program” following a vehicle attack in New York on Tuesday that left eight people dead.

“I have just ordered homeland security to step up our already extreme vetting program. Being politically correct is fine, but not for this!” the president wrote on Twitter, hours after a man driving a pickup truck ploughed through a bike path in lower Manhattan in what has been treated as an “act of terror”.

The victims include a Belgian citizen and five Argentinians.

Trump ran for office in 2016 pledging to crack down on illegal immigration for security reasons, and has called for “extreme vetting” of foreigners entering the US. He did not specify how or when practices would change.

Officials speaking on condition of anonymity identified the attacker as 29-year-old Sayfullo Saipov, an Uzbek citizen who came to the US legally in 2010. Saipov has a Florida driving license but may have been living in New Jersey, reports said. The suspect was shot by police and is in custody. He underwent surgery and was in critical condition but was expected to survive.

Witnesses have said the suspect was shouting “Allahu Akbar”, Arabic for “God is great”, during the attack, and there have been multiple – unconfirmed – reports that a note expressing support for Isis has been found in the rented truck used in the attack.

The White House said Trump was briefed on the attack by his chief of staff, John Kelly.

In initial comments, Trump wrote 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.!”

Eight people killed in New York 'act of terror' after truck drives on to bike path Read more

The tweet came roughly 30 minutes after the first lady, Melania Trump, responded to the attack on Twitter by saying: “My heart breaks for #NYC today. Thoughts & prayers as we monitor the situation.”

Trump later tweeted: “We must not allow ISIS to return, or enter, our country after defeating them in the Middle East and elsewhere. Enough!”

Although there is an Isis presence in Uzbekistan, it is not one of the countries subject to Trump’s controversial travel ban.

Roughly 90 minutes after Trump’s first tweet on the attack, he expressed his condolences for the victims. “My thoughts, condolences and prayers to the victims and families of the New York City terrorist attack. God and your country are with you!” Trump said on Twitter.

Play Video 2:22 Eight people killed after truck drives on to New York bike path – video report

The New York City mayor, Bill de Blasio, and the New York governor, Andrew Cuomo, condemned the attack as an act of terrorism, but urged New Yorkers not to be afraid of further attacks.

De Blasio told reporters: “The bottom line is we are going to go about our business. We are not going to be deterred.”

Cuomo echoed him in more vigorous terms, saying: “We go forward together. And we go forward stronger than ever. We’re not going to let them win.”

In Washington, the Senate minority leader, Chuck Schumer, issued a statement after being briefed by the FBI about the attack. The New York Democrat said: “First and foremost, our thoughts are with the victims and their families. As one of thousands of New Yorkers who regularly rides on the path where this attack took place, I’m particularly grateful to the NYPD and first responders who work day in and day out to keep us safe, and to those that responded to the scene today.

“As the investigation unfolds, it’s critical that we learn what we can from this incident and do everything we can to prevent this from happening again. The scourge of terrorism is unfortunately still with us, and we must remain vigilant as ever.”

Schumer’s words of support were echoed by Speaker Paul Ryan, who tweeted: “Our hearts are with the victims of this apparent act of terror in New York City. Thank you to all the first responders.”