Israeli forces on Monday advanced from three directions toward the battered and densely populated Hamas stronghold of Gaza City, slowly pushing deeper into the Gaza Strip while battering the territory with airstrikes and renewing evacuation warnings to people and hospitals.

On the fourth day of Israel’s invasion, formations of its troops and armored vehicles approached the city and neighboring population centers from the north, east and south, as shown by eyewitness photos and videos verified by The New York Times, as well as satellite imagery.

The deepest penetration, about 3 miles, paralleled the Mediterranean coast toward the Shati refugee camp on the northern flank of the city. South of the city, Israeli forces reached Salah Al-Din Road, one of the main north-south arteries used by people trying to heed the Israeli call to evacuate northern Gaza. One video showed an armored vehicle there firing on a car, which Palestinians said could discourage people from attempting the trip.

Prime Minister Benjamin Netanyahu and other Israeli leaders have vowed to wipe out Hamas, the armed group that controls Gaza, after it led the Oct. 7 attack on Israel that the government says killed more than 1,400 people and seized more than 200 hostages.

“Israel is fighting the enemies of civilization itself,” Netanyahu said at a news conference Monday with foreign news outlets. He rejected the idea of a cease-fire, backed by humanitarian groups and the United Nations General Assembly, among others, adding, “calls for a cease-fire are calls for Israel to surrender to Hamas, to surrender to terrorism.”

Israel has responded to the Oct. 7 assault with a ferocity far beyond its previous conflicts with Hamas, first with an intensive air and artillery bombardment that is still underway, with hundreds of strikes daily. The campaign has killed more than 8,000 Palestinians in Gaza, according to Hamas authorities, displaced more than 1 million and physically devastated large swathes of the Palestinian territory.

Israel faces growing international criticism for its conduct of the war, including the evacuation directive and strikes on civilian targets that Israel says are used to conceal Hamas troops, command posts, tunnels and supply depots. Hospitals, schools and shelters have also been damaged, and UNRWA, the main U.N. agency that provides humanitarian assistance in Gaza, said that 63 of its employees there have been killed since Oct. 7.

President Joe Biden has not publicly found fault with Israel, but has urged restraint. On Sunday he reiterated support for Israel’s right to protect itself, according to the White House, while underscoring “the need to do so in a manner consistent with international humanitarian law that prioritizes the protection of civilians.”

The chief spokesperson for Israel’s military, Rear Adm. Daniel Hagari, declined to say how many Israeli troops were inside Gaza, or where, but confirmed that a combined force of infantry and armored units, operating with air support, was engaging in “expanded ground operations,” but was moving gradually. He said Hamas gunmen typically gather at “staging sites” before trying to attack Israeli soldiers, “after which we strike them from the air.”

Overnight, he added, “dozens of terrorists were eliminated” after they barricaded themselves inside buildings and attempted to attack the soldiers who were moving in their direction. It was not possible to verify Israel’s account of the fighting.

Hamas’ armed wing released a video showing three women who were kidnapped Oct. 7; one of them sharply criticizes Netanyahu, saying the hostages are being held in “unbearable conditions” and demanding that Netanyahu exchange them for Hamas prisoners held by Israel. Families of hostages have made similar appeals.

Netanyahu’s office called the video “cruel psychological propaganda,” and in his news conference, the prime minister said the invasion of Gaza “actually creates the possibility of getting our hostages out.”

The Israeli military said it had rescued a woman, a soldier, taken hostage in the Hamas incursion and held in Gaza.

Conditions are dire for civilians in Gaza, under an Israeli and Egyptian blockade. Forty-seven trucks carrying food, water, medical supplies and other humanitarian aid entered Gaza through the Rafah crossing with Egypt on Sunday, according to a Palestinian official at the crossing. That was the largest one-day total in the nine days since the shipments began, but less than half of what the U.N. says is needed.

Israeli and Western officials say Hamas has stockpiled ample supplies but is hoarding them for itself rather than putting them to humanitarian use — a charge the group denies. Fuel is scarce in Gaza, leaving most people without electricity and hobbling hospitals and shelters, but Israel has refused to allow fuel into the territory, arguingit would go to Hamas.