Fears of a possible ground invasion intensified among Gazans last night. Israel's Iron Dome anti-missile system knocked out a third rocket launched on Tel Aviv in as many days, and Jerusalem ramped up its rhetoric at home and abroad on the need to deter Hamas.

Earlier Israel had kept up its determined bombardment of Hamas targets for a fourth successive day, destroying the headquarters of the de facto Hamas government, including the office of its Prime Minister Ismail Haniyeh, and flattening a three-storey house in Jabalya identified as being that of the northern commander of the faction's military wing.

Tel Aviv residents on the city's coastal boardwalk, who had earlier scrambled for cover after air-raid sirens sounded, cheered as they watched the intercepting missile from the anti-rocket battery leave a trail of smoke in the wake of the successful strike. The rockets launched at Tel Aviv and at the vicinity of Jerusalem on Friday appear to testify to the improved weapons technology of Gaza militants.

The intensified air strikes came as Egyptian-led attempts to broker a ceasefire and end Israel's four-day-old Gaza offensive gained momentum. Leaders of Hamas and two allies, Qatar and Turkey, were in Cairo for talks with Egyptian officials, and the Arab League was holding an emergency meeting.

The White House said President Barack Obama was also in touch with the Egyptian and Turkish leaders. The United States has solidly backed Israel so far.

The fresh bombings include some 200 strikes in the early hours of yesterday, many of which were clearly audible across Gaza City. They brought the death total to 42 Palestinians including at least 13 civilians since the present escalation started on Wednesday with the targeted strike on the head of Hamas's military wing, Ahmed Jaabari.

Meanwhile, Israel launched a diplomatic public relations campaign emphasising plans to increase the mobilisation of reservists from 16,000 to 30,000 which the Israeli news service Ynet depicted as meaning "to lay the ground for an expanded offensive on Gaza". The news service said the IDF was geared towards a significant expansion of its operations in Gaza, and showed harrowing images of a baby injured in the rocket attack which killed three Israelis in the town of Kiryat Malachi on Thursday.

Israel's notably hardline foreign minister, Avigdor Lieberman, said yesterday the military offensive had three main goals: to restore calm in Israel's south, reinstate Israel's deterrence and to destroy the stockpiles of long-range missiles belonging to Gaza terror groups. He added: "The only way we can achieve peace and security is to create real deterrence via a crushing response that will make sure they don't try to test us again. This isn't an all-out war, but an operation with defined goals."

He stressed that if a ground invasion was launched, Israel would have to see it through, adding that this had not been done during the three-week Operation Cast Lead in the winter of 2008-9. This, insisted Mr Lieberman, "was why we failed to achieve our goal and had to pay a heavy toll when it came to global public opinion".

The IDF had said earlier that it had distributed 200,000 leaflets and issued 12,000 text messages to Gaza residents warning them to stay away from Hamas facilities and personnel – which presumes they know where they are. Earlier there had been a large fire around the police headquarters in Gaza City after strikes on two small security missiles and the Hamas police headquarters. No one was in the buildings, which, like almost all other Hamas offices, have been evacuated since the escalation began. The Hamas interior ministry said that, as devout Muslims streamed to the area for early-morning prayers, a government compound was also hit, but did not report any casualties.

Residents living beside the Council of Ministers building, which has housed government offices since Hamas seized full control of the Gaza Strip in 2007, showed windows smashed by the shock of the 5am blast that demolished the HQ. Hamad Abdullah, 38, who lives in a building opposite, said the blast had knocked over cupboards and sent a television crashing to the floor. He added: "We weren't worried because this is not a military place. Many leaders have been received here, including Jimmy Carter." He insisted that Mr Haniyeh had even received the Egyptian Prime Minister in his office in the building on Friday. He said that residents of a building on the other side of the office had evacuated it yesterday, but he had not considered moving. "I have nowhere to go," he added.

Another resident Ahmed Hatem, 60, said he had been worried about living so close to the de facto government HQ but still condemned the attack as "barbaric". He said: "The building was used for civilian administration and for the Palestinian people." He was adamant that Hamas should continue "resisting" adding: "If an enemy attacks your country, you would fight back."

At the destroyed Jabalya refugee camp building, where concertinaed concrete slabs and twisted reinforcing cable testified to the force of the bomb that struck, it was far from clear that anyone had been in the house at the time, though medics said that it had caused up to 27 injuries in nearby homes. The IDF said it had been targeting ammunition for use by militants under the building. Although there were shouts of "resistance" from some local youths as he spoke, Muntassa Qamoud, 20, who lived in the partially destroyed building next door, said: "After this experience I want this fighting to stop."

Gershon Baskin, a liberal Israeli who helped to broker the deal under which Hamas released the abducted Israeli sergeant Gilad Shalit, said last week that a draft deal to end the previous week's flare-up of violence was already circulating among the parties when Israel launched its strike on Ahmed Jaabai and other targets on Wednesday.

A leading Gaza independent analyst, Mukhama Abu Sada, professor of political science at Al Azhar University, said militants' use of long-range rockets would not be tolerated by Israel, which meant the conflict could go on for "a long time". It had three options. First was to launch a ground invasion to remove Hamas "completely from power". This would not only cause regional instability but could also leave Gaza in a state of anarchy, allowing other more militant factions that have in the past been contained by Hamas to continue with attacks. Second was in his view the likeliest: "to teach Hamas a lesson". Third was to reoccupy Gaza.

He said: "Those who support Hamas are very happy that they launched rockets at Tel Aviv and Jerusalem. But others are fed up and exhausted and say 'we just want to live a normal life'. There is no military solution to the problem of Gaza. Even if you kill all the Hamas leadership it will not put an end to the Palestinian struggle. There has to be a political solution."

A mother's story

Perhaps because she was still so traumatised as she lay on her back in the orthopaedic wing of northern Gaza's Kamal Adwan Hospital, Maha, a 38-year-old mother of three, didn't want to give her surname. It was the first day of the current escalation of Gaza violence, and the first bombing after the one that killed the leader of Hamas's military wing, Ahmed Jaabari. It left her with a broken hip and totally destroyed her house in Beit Lahiya.

In one sense, Maha is a statistic, for she is one of a large number of injured civilians. By Friday, according to the independent Palestinian Centre for Human Rights, 253 of the 257 wounded in more than 600 air strikes have been civilians; four were militants. Of the 253, the centre says, 62 were aged under 18, and 42 were women.

With her leg resting on a stabilising metal brace, Maha explained that Israel's defence forces had not been targeting her house but an open field behind it, which they may have had reason to consider was being used by militants. "The whole neighbourhood seemed to know there was a tunnel under the field. But we did not know."

She went on: "I was sitting in my house with my three girls, and my sister was visiting. We were all in one room together. Something hit the house. There we were all together, and the next second we were thrown apart. I was lying on the floor and a wall fell on top of me. As I was under the rubble, I started screaming and screaming – I thought 'this is it'… Nobody could hear me because of the sound of the bomb. It was all so loud nobody could hear anyone. While this was happening, my five-year-old daughter was hiding in the wreckage and was completely fine.

"My sister told the medics I was under the rubble. Otherwise they may not have found me. I thought we had all been buried under the rubble, but my children and my sister are all fine – one daughter, who is 18, has a broken arm and bruising, but really we are fine. All of a sudden, I felt a huge weight lifted off my body – they found me and brought me here."

She drew breath, and paused for a moment. "I want this fighting to stop. I have lived through the 2008-09 three-week invasion of Gaza [Operation Cast Lead] which was an experience even worse than this. I do not want any more of my loved ones or anyone else's loved ones to die or get hurt."

A troubled history

January 2009 Israeli tanks enter Gaza in ground war against Hamas after a week of air strikes. It lasts 22 days and leaves more than 1,200 Palestinians and 13 Israelis dead.

April 2009 Palestinian attacks Israeli children, killing 13-year-old boy.

December 2009 Rabbi and father of seven killed in West Bank drive-by shooting.

January 2010 Israeli air strikes on Gaza weapons tunnels kill three.

February 2010 Israeli woman murdered by two Palestinians at monastery.

May 2010 At least 19 killed when Israel attacks aid-carrying ships.

June 2010 Israeli troops attack worshippers at Al-Aqsa mosque. Two months later, Hamas shoot four dead, including two women.

March 2011 Israeli family of six murdered in their beds. Subsequent Israeli attack on Gaza kills eight. Jerusalem bus station bomb kills one and injures 39. Israeli planes bomb Gaza.

May 2011 Thousands of Palestinians fired at as they gather along border to demand right to return to homes, now on Jewish land. Israel accused of killing 13 Palestinian refugees.

August 2011 Egyptians and Palestinians kill eight Israelis.

March 2012 Gaza militants launch over 300 rockets, missiles and shells into southern Israel, wounding 23 civilians. Israel retaliates with air strikes on Gaza, killing 22 militants.

September 2012 Militants open fire on Israeli soldiers and civilians workers on Israel-Egypt border.

October 2012 Israeli forces kill seven Palestinians.