“There are not and will not be limits for this war,” Bennett said.
On Saturday, the army said security forces had launched the operation in the city of Jenin, in the north of the occupied West Bank, its adjacent refugee camp and nearby villages.
The Palestinian health ministry said at least one Palestinian was killed by Israeli gunfire, while the Red Crescent said 12 others were wounded.
The Islamic Jihad militant group said that one of its fighters, whom it identified as 23-year-old Ahmed al-Saadi, died during clashes in the camp.
A crowd of mourners marched through the streets carrying Saadi’s body on a stretcher covered with the group’s flag.
Palestinian security sources said part of Saturday’s operation was to identify the home of the Tel Aviv assailant ahead of demolishing it.
Human rights activists have repeatedly denounced Israel’s policy of demolishing the homes of Palestinian attackers, saying that those affected are often relatives or neighbours unconnected to the crime. Israel says it acts as a deterrent.
The Israeli army said troops and border police were “conducting counterterrorism activity” in the Jenin area, when gunmen had opened fire “endangering their lives.”
In response, troops opened fire “towards the armed assailants”, the army said, adding there were no casualties in Israeli ranks.
“An M16 assault rifle used by an assailant to attack the troops was confiscated.”
The Jenin refugee camp is a stronghold of armed factions, where three other Palestinians linked to an anti-Israeli attack were killed by the army last week.
Saturday’s raid comes a day after Israel said it had killed Raad Hazem, 28, the alleged Tel Aviv attacker.
In addition to giving security forces free rein to curb a surge in violence, Bennett on Friday ordered the closure of the Jalameh checkpoint between the Jenin area and Israel.
On Saturday evening, the Israeli defence ministry body responsible for civil affairs in the Palestinian territories (COGAT) announced the closure of a second checkpoint in the area and an “intensification” of checks.
“We will do whatever it takes, whatever is necessary, for however long and wherever needed, until both safety and the sense of security are restored,” army chief Lieutenant General Aviv Kochavi told soldiers in the West Bank, in a video released by the military.
On Friday, the father of the Tel Aviv attacker, Fathi Hazem—a retired Palestinian security forces officer according to Palestinian sources—struck a defiant tone.
Speaking to hundreds at the family home in Jenin, he said the Palestinian people were looking for “freedom and independence”.
A total of 14 people have been killed in attacks in Israel since 22 March, including some carried out by assailants linked to or inspired by the Islamic State group.
Over the same period, at least 10 Palestinians have been killed, including assailants.
Islamic Jihad and fellow Islamist movement Hamas praised the Tel Aviv attack—drawing criticism from the UN—but did not claim responsibility.
Palestinian leader Mahmud Abbas denounced the Tel Aviv attack, while the United States stressed anew its support of key ally Israel.
The Tel Aviv attack killed three Israeli men: childhood friends Tomer Morad and Eytam Magini, as well as father of three Barak Lufan.
It came amid heightened tensions during Ramadan, after violence flared during the Muslim holy month last year leading to 11 days of devastating conflict between Israel and Palestinian militants in the Hamas-ruled Gaza Strip.
Earlier this month, Israeli security forces killed three Islamic Jihad militants when they came under fire during an operation to arrest them in Jenin.
The raid, in which four Israeli soldiers were wounded, followed another deadly attack on 29 March in Bnei Brak, a largely ultra-Orthodox Jewish city near Tel Aviv.
The Palestinian assailant, who had also come from Jenin, shot dead two Israeli civilians, two Ukrainians and an Arab-Israeli policeman.