Gazan authorities reported four Palestinians killed in Israeli strikes in the escalation that began Saturday, but Israel disputed their account of the deaths of a pregnant mother and her baby.
One 58-year-old Israeli man was killed by a rocket strike on the city of Ashkelon near the Gaza border, Israeli police and the hospital said.
The latest flare-up came with Hamas, the Islamist movement that rules the blockaded Gaza Strip, seeking further concessions from Israel under the ceasefire.
Israel said 430 rockets had been fired from the Palestinian enclave since Saturday and its air defences intercepted many of them.
Beyond the man killed, an 80-year-old woman was seriously injured in a rocket strike on the Israeli city of Kiryat Gat, police and medics said.
A man was also hospitalised in Ashkelon, said police, which spoke of other injuries without providing details.
A house near Ashkelon was damaged while other rockets hit open areas.
The Israeli army said its tanks and planes hit some 200 militant targets in Gaza in response.
They included an Islamic Jihad attack tunnel that stretched from southern Gaza into Israeli territory, it said.
Two multi-storey buildings in Gaza City were destroyed.
Israel said one of the buildings included Hamas military intelligence and security offices.
Turkey said an office for its state news agency Anadolu was located in the building and strongly denounced the strike.
Israel said the other building housed Hamas and Islamic Jihad offices.
The Gaza health ministry said Israeli strikes killed a 14-month-old baby and her pregnant mother, 37, in addition to two Palestinian men, while 40 were wounded.
Israeli army spokesman Avichay Adraee on Twitter challenged the account of the mother and her baby being killed in an Israeli strike, suggesting they died from Palestinian fire.
Adraee did not provide more details and the army refused to comment further. Israel's military said it was targeting only military-related sites.
- Video threat -
As the exchange of fire continued, Israeli Prime Minister Benjamin Netanyahu held consultations with security chiefs on Saturday but had not commented publicly.
A statement from Hamas ally Islamic Jihad claimed responsibility for at least some of the rocket fire and said it was prepared for more.
Its armed wing distributed a video showing militants handling rockets and threatening key Israeli sites, including Ben-Gurion international airport near Tel Aviv.
Israel closed its people and goods crossings with Gaza as well as the zone it allows for fishermen off the enclave until further notice due to the rocket fire.
Egyptian and UN officials were engaged in discussions to calm the situation, as they have done repeatedly in the past, while the European Union called for an immediate halt to rocket fire from Gaza.
The UN envoy for the Israeli-Palestinian conflict, Nickolay Mladenov, called on "all parties to immediately de-escalate and return to the understandings of the past few months."
The United States condemned the rocket attacks on Israel and said it fully supported its "right to self-defence against these abhorrent attacks."
- Visit to Cairo -
The escalation follows what had been the most violent clashes along the Gaza border in weeks on Friday.
Four Palestinians, including two Hamas militants, were killed after two Israeli soldiers were wounded in a shooting during weekly protests there.
Israel and Palestinian militants in the Gaza Strip have fought three wars since 2008 and fears remain of a fourth.
A ceasefire between Israel and Hamas brokered by Egypt and the United Nations had led to relative calm around Israel's April 9 general election.
But the past week saw a gradual uptick in violence.
With the ceasefire at risk, a Hamas delegation led by its Gaza head Yahya Sinwar went to Cairo on Thursday for talks with Egyptian officials.
The ceasefire has seen Israel allow Qatar to provide millions of dollars in aid to Gaza to pay salaries and to finance fuel purchases to ease a severe electricity shortage.
- Eurovision looms -
Several factors may lead Israel to seek to calm the situation.
Netanyahu is engaged in tough negotiations to form a new government following last month's election, while Israel is due to host the Eurovision song contest in Tel Aviv from May 14-18.
The country also celebrates its Independence Day on Thursday.
On the Gazan side, the Muslim holy month of Ramadan begins in the week ahead.
Palestinians have participated in regular demonstrations and clashes along the Gaza border for more than a year, calling on Israel to ease its crippling blockade of the enclave.
At least 271 Palestinians have been killed by Israeli fire since the protests began in March 2018, the majority along the border.
Two Israeli soldiers have been killed in that period.
Israel accuses Hamas of using the protests as cover to carry out attacks and says its actions are necessary to defend the border and stop infiltrations.