DEIR AL-BALAH, Gaza Strip — Israeli troops battled Hamas and attacked underground compounds on Tuesday with a focus on northern Gaza, from which an estimated 800,000 Palestinians have fled south despite continued Israeli bombardment across the besieged enclave.

Buoyed by the first successful rescue of a captive held by Hamas, Prime Minister Netanyahu has rejected calls for a cease-fire and again vowed to crush Hamas’ ability to govern Gaza or threaten Israel following its bloody October 7 rampage, which ignited the war.

The war has also threatened to ignite even heavier fighting on other fronts. Israel and Lebanon’s Hezbollah have traded fire on a daily basis along the border, and Israel and America have struck targets in Syria linked to Iran, which supports Hamas, Hezbollah and other terrorist groups in the region.

The military said it shot down a drone outside Israeli airspace on Tuesday near the Red Sea city of Eilat, without providing further details. Earlier this month, an American Navy destroyer in the Red Sea intercepted three cruise missiles and several drones launched toward Israel by Iran-backed Houthi rebels in Yemen.

In the West Bank, where Palestinian-Arab violence has also surged, the army demolished the family home of a senior Hamas official, Saleh al-Arouri, exiled over a decade ago. The head of the local council in the village of Aroura, Ali Kaseeb, said the home had been vacant for 15 years.

An Israeli military spokesman, Jonathan Conricus, said ground operations in Gaza are focused on the north, including Gaza City, which he said was the “center of gravity of Hamas.”

“But we also continue to strike in other parts of Gaza. We are hunting their commanders, we are attacking their infrastructure, and whenever there is an important target that is related to Hamas, we strike it,” Mr. Conricus said.

The military said it struck some 300 militant targets over the past day, including compounds inside tunnels, and that troops had engaged in several battles with Palestinian militants armed with antitank missiles and machine guns.

Larger ground operations have been launched both north and east of Gaza City, which before the war was home to over 650,000 people.

Video footage released by the military showed soldiers walking across an open area as heavy gunfire echoes in the background and setting up a position in the ruins of a heavily damaged building.

Mr. Conricus said some 800,000 people have heeded the Israeli military’s orders to flee from the northern part of the strip to the south. But tens of thousands of people remain in and around Gaza City, and casualties are expected to mount on both sides as the battle moves into dense, residential neighborhoods.

In a news conference late Monday, Mr. Netanyahu rejected calls for a cease-fire to facilitate the release of captives or end the war, which he has said will be long and difficult. “Calls for a cease-fire are calls for Israel to surrender to Hamas,” he told a news conference. “That will not happen.”

Mr. Netanyahu, who faces mounting anger over Israel’s failure to prevent the worst surprise attack on the country in a half century, also said he had no plans to resign.