GAZA, July 8 – The Palestinian prime minister from Hamas, Ismail Haniya, on Saturday called for a mutual cease-fire with Israel after Israeli forces pulled out of most of the northern Gaza Strip, perhaps responding to Israeli hints that a package deal might be possible to end a military and political crisis.

Eleven days after Israeli forces entered Gaza to free an abducted Israeli soldier and stop rocket fire by militants, they continued to hold territory in northern Gaza, east of Beit Hanun and the Erez crossing, and in the south, near Rafah. Early Saturday, the military also moved into eastern Gaza near the Karni crossing with Israel, where there were fierce clashes with militants on the eastern edge of Gaza City.

An Israeli official said in response to Mr. Haniya that there would be no cease-fire until the release of the Israeli soldier, Cpl. Gilad Shalit.

The violence continued Saturday evening, when an Israeli airstrike near Karni killed four Palestinians, including three members of the Hajaj family, who were sitting in their garden. Among the dead were Rawan Hajaj, 6, her brother Muhammad, 20, and their mother, Um Ayman Hajaj. The Israeli Army said it had been aiming at “a cell of armed gunmen” walking in the area and was checking to see whether a missile fired at them might have been responsible.

In fighting earlier in the day, another four Palestinians died, bringing the death toll to at least 44 Palestinians and one Israeli since Thursday. Nearly 20 Palestinians were wounded Saturday, including a 1-year-old, bringing the total of wounded to more than 130, according to Dr. Jumaa al-Saqqa of Al Shifa hospital, Gaza’s largest.

In Israel, a cabinet minister, Ophir Pines-Paz of the Labor Party, suggested that the crisis could end through a package deal of the kind that Egypt had been trying to promote – the release of Corporal Shalit, coupled with a more formal cease-fire and an Israeli promise to release a number of Palestinian prisoners as the cease-fire is shown to have taken hold.

“We have a great interest in changing the rules of the game,” said Mr. Pines-Paz, who is also a member of Israel’s inner security cabinet. “If we reach a situation in which there are no kidnappings, no rockets, no tunnels, no raids into our territory, certainly Israel will have to reciprocate,” he told Israel Radio.

On Friday, Avi Dichter, the minister of internal security, said that once the soldier was released, “if Israel will be obliged to in the framework of this agreement, within the calm that will be imposed on the ground by the Palestinian Authority, to release Palestinian prisoners as a gesture to this calm, Israel knows how to do that.”

“In fact,” he said, “Israel has done it before in the past.”

Prime Minister Ehud Olmert has ruled out a prisoner trade, and Israel is eager to try to deter kidnappings by showing that it will not bargain for the corporal with his militant captors. But Mr. Olmert had promised to release prisoners to the Palestinian president, Mahmoud Abbas, at their first formal summit meeting, Mr. Abbas said Friday. Former Prime Minister Ariel Sharon did something similar when the two leaders met in February 2005.

In his statement, Mr. Haniya said, “In order to get out of the current crisis, it is necessary that all parties restore calm on the basis of mutually stopping all military operations,” and he called for the renewal of talks.

His spokesman, Ghazi Hamad, said: “We asked for a mutual cease-fire and a return to the place of calmness. Israeli forces should pull out of the Gaza Strip and stop their aggression.”

An Israeli military spokesman said the army’s objectives in northern Gaza had been met, even though the number of Qassam rockets hitting Israel had increased, with 14 launched Friday. Capt. Noa Meir said the Israelis “are in an ongoing operation, and we’re not ruling out going back into the north again.” She said Israel had damaged “the terrorist infrastructure, the rocket-launching infrastructure and put general pressure on the Hamas government to release Corporal Shalit.”

The Israelis say they also destroyed Qassam rocket manufacturing sites and storehouses.

Israeli troops moved in through the Karni crossing to look for tunnels and explosives, the army said, with engineering troops supported by armor. Early Saturday morning, the Israelis blew up a tunnel near Karni from the air, Captain Meir said. Later on Saturday, there were some serious clashes with Palestinian militiamen, with exchanges of machine-gun fire and some helicopter-fired missiles.

Palestinian militants in the area, a Hamas stronghold known as Shijaia, were seen planting explosives along streets and guarding street corners.

Overhead was the steady buzzing of a missile-armed Israeli drone aircraft, which the Palestinians here call the zananah, a made-up word for the aircraft’s buzzing sound. Late Saturday, the Israelis pulled back through the crossing.

In Beit Lahiya, the scene of the worst fighting so far, residents cleaned up their houses on Saturday after the Israeli troops departed. The troops began moved out at about 1 a.m. and were mostly back into Israel by 9 a.m. They left behind them broken walls, smashed houses and angry Palestinians.

In the Atatrah neighborhood, Muhammad Erheem, shaken, showed a reporter through his large house on a hill, which 25 Israeli soldiers commandeered on Thursday night before leaving Saturday morning. They locked the 13 members of the Erheem family in a single storeroom, letting them in small groups, under guard, only to use the toilet, Mr. Erheem said.

The troops moved into the compound behind an armored bulldozer that crushed two smaller buildings – a stable and a water pump house – and also killed a number of prize goats, who were buried alive in the sand as the bulldozer passed over.

“I could hear my goats screaming above the noise,” Mr. Erheem said. “Then they banged on the door and I opened it, and 10 soldiers came in, very agitated, and ordered me to strip, and one cocked his pistol at me and shouted, ‘Where is the weapon?’ I speak fluent Hebrew, thank God, and I said I had no weapon and there was no weapon in the house.”

The soldiers made holes in the walls to establish sniper positions, he said, and dismantled furniture, propping beds up against walls to help protect them from any incoming fire and “using my house like a hotel.”

But Mr. Erheem was shocked that the soldiers brought wine into his house, and that one of them stripped down to his underwear in the heat and walked around that way in front of the women. Mr. Erheem also pointed out a Koran that had been ripped apart and scattered.

His son, Bakr Erheem, 25, had been married five days before, and there was a big party, the house still full of wedding bouquets and unopened gifts. Their wedding bed was dismantled and broken, holes tore through the concrete walls. Outside, his two water-tanker trucks – he had a business selling filtered water – were smashed, their engines pushed into the driver’s cab.

Mr. Erheem said one soldier took a gold necklace, and he showed a purse he said had had gold jewelry in it. The soldiers had replaced the gold, he charged, with a gold-colored piece of metal that looked to be part of an antenna.

The Israeli Army said that troops had regulations about requisitioning houses and that it would investigate all allegations of wrongdoing.