The Palestinian Ministry of Health in Gaza identified the pregnant woman as 23-year-old Inas Khamash, and her 18-month-old daughter as Bayan Khamash.
On Wednesday night and Thursday, Israeli aircraft struck more than 150 targets in Gaza and Palestinian militants fired scores of rockets including a long-range missile deep into Israel, escalating fighting despite the ongoing truce talks.
At least three Palestinians were killed in Israeli airstrikes, while seven people were injured by rockets fired into the Jewish state.
IDF said demonstrators were throwing rocks and burning tires at Israeli troops, causing them to respond with live fire "in accordance with the rules of engagement".
While Israeli officials insisted that their large-scale attack on Gaza was in response to Hamas rocket fire, Haaretz reports that it was in fact the Israeli army that sparked the escalation by killing two Palestinians with tank fire, after they allegedly mistook a training exercise for an actual attack.
"The principle of returning [captured] soldiers, alive, wounded or dead - is a principle we all grew up on, and must not allow to be lost", he said.
"This is a blatant crime and Israel is responsible for it", he added. Those have hit a stumbling block as Hamas is insisting that the release include all prisoners who were part of a 2011 prisoner exchange for Israeli soldier Gilad Shalit but have since been rearrested, according to Palestinian officials, who declined to be named because of the sensitivity of the talks.
In the Gaza Strip, power generation is so paltry that, even with imports from Israel and Egypt, it gets just one-third of what it needs - so the 2 million Gazans struggle on with an average of just four hours of electricity a day.
i24NEWS Defense Correspondent Daniel Tsemach contributed reporting from southern Israel.
Rocket alert sirens sounded in Beersheba, a city of 200,000 inhabitants some 40 km from the Gaza Strip, for the first time since the last war in 2014. Rockets and mortar shells also landed in open areas, in addition to the 25 projectiles that were headed to densely populated areas and were intercepted by the Iron Dome aerial defense system.
"We urge the Palestinians and the Israelis to display restraint and to take measures towards ruling out the relapses of violence with unpredictable aftermaths", Zakharova said.
The Israeli security cabinet and the Hamas leadership convened separately late on Thursday, with the truce offer brokered by Egypt and the United Nations on the table.
As Israeli airstrikes persist and the living situation gravely worsens, Palestinians in Gaza are on the brink of a full-blown humanitarian crisis due to Israel's 10-year siege.
"If the current escalation however is not contained immediately, the situation can rapidly deteriorate with devastating consequences for all people", Mladenov said. The headline also is flawed by a murky portrayal of events: "Clashes Between Israel and Gaza".
The Gaza Strip, home to almost two million Palestinians, has been reeling under a crippling Israeli blockade since 2007.