The clash, in which Israeli television said five gunmen were killed and the Islamist Hamas movement said it had killed 10 Israeli soldiers, appeared to wreck international hopes of turning a brief lull in fighting into a longer-term ceasefire.
After the infiltration at Nahal Oz, a kibbutz collective village due east of Gaza City, the Israeli army warned thousands of Palestinians to flee their homes in areas around Gaza City. Such warnings usually precede retaliatory strikes.
As night fell over Gaza, army flares illuminated the sky and the sound of intense shelling could be heard.
The incident was not the only breach of the fragile truce. Eight children and two adults were killed by a blast at a park in northern Gaza and four Israelis were reported to have been killed by cross-border Palestinian mortar fire.
Residents blamed the park explosion, which also wounded 40 adults, on an Israeli airstrike, but Israel said a misfiring rocket launched by Hamas militants had hit the public garden in the Beach refugee camp.
Israeli media said four Israelis were killed by a mortar round fired out of Gaza in a separate incident. The military declined immediate comment.
Israeli forces had said they were firing only when fired upon while army engineers hunted infiltrator tunnels from the Gaza Strip's eastern frontier. They accused Palestinians of launching at least 17 rockets across the border.
Gaza's dominant Hamas Islamists had called for a pause in hostilities on Monday, the 21st day of their conflict with Israel, to coincide with Eid, which marks the end of the fasting month of Ramadan.
Israel initially balked, having abandoned its own offer to extend a 12-hour truce from Saturday as Palestinian rockets kept flying. However, calm gradually descended through the night with just the occasional exchange of fire heard until a series of blasts shook Gaza in the afternoon.
Pools of blood lay on the ground in the Beach refugee camp garden in the aftermath of one of the blasts.
"We came out of the mosque when I saw the children playing with their toy guns. Seconds later a missile landed," said Munther Al-Derbi, a resident of the camp.
"May God punish ... Netanyahu," he said, referring to Israeli Prime Minister Benjamin Netanyahu.
At roughly the same time, another blast shook the grounds of Gaza's main Shifa hospital, without causing any casualties. Israel, which has previously accused Hamas fighters of hiding in the hospital, again blamed an errant militant missile.
PRESSURE Foreign pressure on Netanyahu mounted on Sunday, with both U.S. President Barack Obama and the U.N. Security Council urging an immediate ceasefire that would allow relief to reach Gaza's 1.8 million Palestinians, followed by negotiations on a more durable cessation of hostilities. Israel wants guarantees Hamas will be stripped of its tunnels and rocket stocks. It worries the Palestinian Islamists will parlay the truce talks mediated by their friends in Qatar and Turkey into an easing of an Israeli-Egypt blockade on Gaza. Speaking to U.N. Secretary-General Ban ki-Moon on Monday before news of the border infiltration was announced, Netanyahu accused the Security Council of siding with Hamas. "The statement ... relates to the needs of a murderous terrorist group that attacks Israeli civilians, and has no answer for Israel's security needs - among them a demilitarization of the Gaza Strip," Netanyahu said. "Israel accepted three U.N. proposals for humanitarian truces and Hamas violated them all," his office quoted him as saying. Some 1,054 Gazans, most of them civilians, have died in the three-week-old conflict. Israel has lost 43 soldiers to Gaza fighting and another three civilians have been killed by Palestinian shelling.
In New York, Ban deplored what he described as a lack of resolve among all parties in the conflict.
"It's a matter of their political will. They have to show their humanity as leaders, both Israeli and Palestinian," he told reporters. "Why these leaders are making their people to be killed by others? It's not responsible, (it's) morally wrong."
Israeli officials were more reticent about Sunday's phone conversation between Obama and Netanyahu in which the U.S. president appeared to link any Gaza demilitarization to a peace accord with the Palestinians that is nowhere on the horizon.
SHELLING
Israel signaled it would prefer a de facto halt to fighting rather than an agreement that would preserve Hamas's arsenals and shore up its status by improving Gaza's crippled economy.
"The situation now is an unlimited truce," Israel's chief military spokesman, Brigadier General Motti Almoz, told Israel Radio on Monday. "The IDF (Israeli Defence Forces) is free to attack after any fire if there is any."
Some residents in Gaza reported they had received a recorded telephone message on Monday which said in Arabic: "Listen Hamas, if you are still alive, you should know that if you continue, we will respond, we will respond violently."
In another attempt at psychological warfare, Israel dropped leaflets over Gaza listing dozens of names of gunmen from Hamas and its ally, Islamic Jihad, that the military says it has killed since the start of the offensive.
An opinion poll broadcast by Channel 10 TV showed overwhelming Israeli public support for continuing the Gaza offensive until Hamas is "disarmed".
Deputy Islamic Jihad chief Zeyad Al-Nakhala said mediation had made progress and the group was working with neighboring Egypt to craft a deal.
"We are days away from the end of the battle, the clouds will clear and you (Palestinians) will see victory," he told Islamic Jihad's radio station Al-Quds, "We will not accept anything less than ending the blockade."
U.S. Secretary of State John Kerry visited the region last week to try to stem the bloodshed, his contacts with Hamas - which Washington formally shuns - facilitated by Egypt, Turkey, Qatar and Western-backed Palestinian President Mahmoud Abbas. Tension between Netanyahu's government and Washington has flared over U.S. mediation efforts, adding yet another chapter to the prickly relations between the Israeli leader and Obama.
Repeated U.S.-led negotiations over 20 years have failed to broker a permanent deal. The most recent round collapsed in April, with Palestinians livid over Jewish settlement building in the occupied West Bank and Israelis furious that Abbas had signed a unity pact with old foe Hamas. Qatari Foreign Minister Khaled Al-Atteya said Israel had not respected an ceasefire agreement brokered by Cairo that ended the last Gaza war in 2012 and it was time the blockade of the coastal enclave - also enforced by next-door Egypt - was lifted.
"We have worked with the U.S secretary of state and we were about to achieve substantial results, and the brothers in Hamas acted positively, but the one who rejected the Kerry proposal was Israel," Al Atteya told Al-Jazeera TV.
The main U.N. agency in Gaza, UNRWA, said more than 167,000 displaced Palestinians had taken shelter in its schools and buildings, following repeated calls by Israel for civilians to evacuate whole neighbourhoods ahead of military operations.
(Additional reporting by Amena Bakr in Doha; Writing by Maayan Lubell and Dan Williams; Editing by Crispian Balmer and Paul Taylor)
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