The Palestinian death toll in an Israeli offensive in the Gaza Strip rose above 500 yesterday as the army said it had killed 10 militants who tunnelled into Israel, while Gazan officials said an Israeli tank shelled a hospital, killing civilians.

A day after he was caught by an open microphone saying sarcastically that the Israeli assault was “a hell of a pinpoint operation”, US Secretary of State John Kerry flew to Cairo to try to secure an end to the two-week conflict.

Despite a UN Security Council appeal on Sunday for an immediate ceasefire in the worst bout of Palestinian-Israeli violence for more than five years, neither the Palestinian Islamist group Hamas nor Israel appeared ready to stop fighting.

Hamas, which killed 13 Israeli soldiers in Gaza on Sunday in the biggest one-day toll for eight years, continued to fire rockets deep into Israel and to dispatch infiltrators.

Israeli jets, tanks and artillery constantly pounded the densely-populated coastal strip, killing 28 members of a single family at the southern end.

At Al-Aqsa hospital in the central Gaza Strip, four people were killed and 70 wounded when an Israeli tank shell slammed into the third floor, housing operating theatres and an intensive care unit, the Health Ministry said.

The Israeli military, which has accused Hamas militants of firing rockets from the grounds of Gaza hospitals and seeking refuge there, had no immediate comment.

Non-stop attacks lifted the Palestinian death toll to 518, including almost 100 children, since fighting started on July 8, Gaza health officials said. Israel says 18 of its soldiers have also died along with two civilians.

Hamas announced late on Sunday it had captured an Israeli soldier in Gaza, displaying a photo ID card and serial number, but there was no confirmation from the Israeli side. The announcement set off rejoicing in the embattled Gaza Strip.

“This is not the time to talk of a ceasefire,” said Gilad Erdan, communications minister and a member of Israeli Prime Minister Benjamin Netanyahu’s inner security cabinet.

“We must complete the mission, and the mission cannot end until the threat of the tunnels is removed,” he told reporters.

For its part, Hamas, weakened by the loss of Egypt and Syria as allies, voiced determination to fight on to break a blockade on Gaza imposed by both Israel and Egypt.

Past conflicts between Israel and its foes in Gaza and Lebanon have usually ended when the United States, the Jewish state’s guardian ally, calls a halt, sometimes hastened by a strike that inflicts high civilian casualties on the Arab side.

While Washington went along with Sunday’s Security Council statement, it has so far defended Israeli actions and refrained from pressuring Netanyahu publicly to stop.

Violence along the Gaza border intensified yesterday and sirens wailed across much of central and southern Israel to warn of rocket attacks.

At least nine missiles were shot down by Israel’s Iron Dome interceptor, the army said.

Looking to take the fight onto Israeli soil, two groups of Palestinian fighters crossed from Gaza via two tunnels in the early morning, opening fire as they entered.

Black and white surveillance footage supplied by the army, showed one group of five or six men crouching and firing in long grass. Seconds later they were hit by a large explosion, which sent a cloud of smoke and debris flying into the air. A military spokeswoman said at least 10 militants died. She did not comment on reports of casualties amongst Israeli forces.

We need to continue to grit our teeth, to shut our ears, to ignore the background noise and to get the job done

Hamas said its men had destroyed an army jeep in the assault. Fighters from Hamas, which controls Gaza, and its allies, have repeatedly tried to infiltrate Israel over the past week through a vast network of hidden tunnels, looking to attack villages and army encampments that dot the border area.

Netanyahu sent in Israeli ground forces on Thursday to destroy the tunnels and the militants’ missile stock pile.

Hamas named the Israeli soldier it claimed to have captured as Shaul Aron, but it did not release any picture of him alive in its hands.

Israel’s ambassador to the United Nations denied the claim, and the Israel’s military said it was still investigating.

“We still cannot rule it out,” military spokesman Lieutenant-Colonel Peter Lerner said on Monday.

A confident-sounding Hamas told Israel to quit Gaza.

“Israel has terribly failed and we advise them to take their soldiers and leave before we kidnap more soldiers in addition to the scores we have already killed and wounded,” said the group’s spokesman, Sami Abu Zuhri.

Any capture of an Israeli soldier would pile pressure on Netanyahu to intensify the military campaign.

He agreed to free more than 1,000 Palestinian prisoners in 2011 to secure the release of a soldier who was held for more than five years by Gaza, and officials have said they want to avoid any repeat of that prolonged drama.

Lerner told reporters that the main focus of fighting remained the Shejaia district, east of Gaza City, where some 72 Palestinians, many of them civilians, were killed on Sunday.

In its push against militants in Shejaia, Israel also suffered its worst losses in the offensive, with 13 soldiers killed – the army’s heaviest one-day loss in battle since 2006.

The high death toll by Israeli standards appeared to cement a public mood of grim determination. Many flags flew at half mast but no leading figures are calling into question the operation.

“We need to continue to grit our teeth, to shut our ears, to ignore the background noise and to get the job done,” columnist Ben Caspit wrote in Ma’ariv. Israel’s army said it had been targeting militants in the clashes, charging that they had fired rockets from Shejaia and built tunnels and command centres there.

The army said it had warned civilians to leave two days earlier.

Sounds of explosions rocked Gaza City through the morning, with residents reporting heavy fighting in Shejaia and the adjacent Zeitoun neighbourhood. Locals also said there was heavy shelling in Beit Hanoun, in the northern Gaza Strip.

“It seems we are heading towards a massacre in Beit Hanoun. They drove us out of our houses with their fire. We carried our kids and ran away,” said Abu Ahmed,who did not want to give his full name for fear of Israeli reprisals.

“It was a night of horror,” the 50 year-old told Reuters.

At the other end of Gaza, medics said 28 members of the Abu Jamea family died when their house was hit by a bomb. Nearby, 10 members of the Seyam family died when they were hit by a tank shell as they tried to flee their house, officials said.

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