Israeli soldiers battled Hamas terrorists in the streets of southern Israel on Sunday and launched retaliation strikes that leveled buildings in Gaza, while in northern Israel a brief exchange of strikes with Lebanon’s Hezbollah militant group raised fears of a broader conflict.

There was still some fighting underway more than 24 hours after an unprecedented surprise attack from Gaza, in which Hamas terrorists, backed by a volley of thousands of rockets, broke through Israel’s security barrier and rampaged through nearby communities.

They took captives back into the coastal Gaza enclave, including women, children and the elderly, whom they will likely try to trade for thousands of Palestinian prisoners held by Israel. Hundreds have been killed on both sides.

Prime Minister Netanyahu said the country was at war and would exact a heavy price from its enemies. Hamas leaders said they were prepared for further escalation.

A major question now was whether Israel would launch a ground assault into Gaza. Mr. Netanyahu vowed that Hamas “will pay an unprecedented price.” But, he warned, “This war will take time. It will be difficult.”

Civilians paid a staggering cost for the violence. Israeli press, citing rescue service officials, said at least 300 people were killed in Israel, including 44 soldiers, while officials in Gaza said 313 people had died in the territory. An Israeli official said the military had killed 400 militants and captured dozens more.

Israeli TV news aired a stream of accounts from the relatives of captive or missing Israelis, who wailed and begged for assistance amid a fog of uncertainty surrounding the fate of their loved ones. In Gaza, residents fled homes near the border to escape Israeli strikes, fleeing deeper inside the territory after warnings in Arabic from the Israeli military.

In neighboring Egypt, a policeman shot dead two Israeli tourists and an Egyptian at a tourist site in Alexandria, the Interior Ministry said. Egypt made peace with Israel decades ago, but anti-Israel sentiment runs high in the country, especially during bouts of Israeli-Palestinian violence.

The flare-up on Israel’s northern border also threatened to draw into the battle Hezbollah, a fierce enemy of Israel’s which is backed by Iran and is estimated to have tens of thousands of rockets at its disposal. Hezbollah fired dozens of rockets and shells on Sunday at three Israeli positions in a disputed area along the border and Israel’s military fired back using armed drones.

Rear Admiral Daniel Hagari, an Israeli military official, told reporters the situation at the northern border was calm after the exchange. But he said fighting was still underway in the south and that there were still hostage situations there.

He said troops had moved into every community near the Gaza frontier, where they planned to evacuate all civilians and scour the area for terrorists.

“We will go through every community until we kill every terrorist that is in Israeli territory,” he said. In Gaza, “every terrorist located in a house, all the commanders in houses, will be hit by Israeli fire. That will continue escalating in the coming hours.”

Hamas said that overnight it had continued to send forces and equipment into “a number of locations” inside Israel. Hamas-linked media reported that the son of Nizar Awadallah, a senior political official, was killed, though the group itself has not reported any senior members being captured, killed or wounded.

The surprise attack Saturday was the deadliest on Israel in decades. In an assault of startling breadth, Hamas gunmen used explosives to break through the border fence enclosing Gaza, then crossed with motorcycles, pickup trucks, paragliders and speed boats on the coast.

They rolled into as many as 22 locations outside the Gaza Strip early Saturday morning, including towns and other communities as far as 15 miles from the Gaza border, while Hamas launched thousands of rockets at Israeli cities.

Israeli media said at least 300 people were killed and 1,500 wounded in Saturday’s attack. Hamas terrorists took an unknown number of civilians and soldiers captive into Gaza, and a line of Israelis with missing relatives snaked outside a police station in central Israel to supply investigators with DNA samples and other means that could help identify their family members.

Israel struck 426 targets in Gaza, its military said, flattening several buildings in giant explosions. The United Nations agency for Palestinians, said more than 20,000 Palestinians left Gaza’s border region to head further inside the territory and take refuge in UN schools.

In a televised address Saturday night, Prime Minister Netanyahu said the military would use all of its strength to destroy Hamas’ capabilities. The Israeli military confirmed that a “substantial” number of Israelis were abducted Saturday without giving an exact figure.

An Egyptian official said that Israel has sought help from Cairo to ensure the safety of the hostages and that Egypt’s intelligence chief had contacted Hamas and the smaller but more radical Islamic Jihad group, which also took part in the incursion, to seek information.

Egypt has often mediated between the two sides in the past. The official said Palestinian leaders claimed that they don’t yet have a “full picture” of hostages, but said those who were brought into Gaza were taken to “secure locations” across the territory.

“It’s clear that they have a big number — several dozens,” said the official, who asked not to be identified because he was not authorized to brief media. Egypt also spoke with both sides about a potential cease-fire, but the official said Israel was not open to a truce “at this stage.”

The shadowy leader of Hamas’ military wing, Mohammed Deif, said the assault, named “Operation Al-Aqsa Storm,” was in response to the 16-year blockade of Gaza, and a series of recent incidents that have brought Israeli-Palestinian tensions to a fever pitch.