While the health ministry said all three Palestinians were killed in Israeli air strikes, local sources have indicated that the Khamashes may have been killed by a misfired Palestinian rocket.
Also. ABC News reported that Israeli combat aircraft struck dozens of targets in the Gaza Strip and three people were reported killed there.
US Envoy Jason Greenblatt who has been involved in peace negotiations between Israel and Palestine, pointed a finger squarely at Hamas for the escalation of violence.
It further claimed that the Israeli missile system Iron Dome had intercepted 25 rockets out of the 150 rockets, which activated sirens in the occupied territories over 125 times.
Residents of Israeli communities near the Gaza border were instructed on Wednesday night to remain near protected spaces.
What outraged the Israeli side is the BBC's "deliberate lying", while not mentioning the fact that the Israeli bombing was triggered by the rockets fired from Gaza toward southern Israel as the Israeli media reported, despite Hamas' denial.
Following May's rocket attacks, Israel and Gaza reached an uneasy cease-fire mediated by Egypt, though rocket launches and airstrike retaliation has continued.
Hamas's military wing said it was behind the rocket attacks.
But the continued outbursts of fire have jeopardized those ceasefire efforts.
In a rapid escalation that began last night, Hamas and other Palestinian movements launched about 180 rockets and mortar shells towards Israeli towns and villages.
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"This is the only barrier of its kind in the world, which will in effect block the possibility of infiltration of Israel from Gaza via the sea", Defense Minister Avigdor Liberman stated at the end of May.
It was the third severe flare-up of violence since July and came despite attempts by United Nations officials and Egypt to secure a long-term truce between Israel and Hamas.
Hamas spokesman Sami Abu Zuhri blamed Israel for the escalation, citing the killing of two Hamas gunmen on Tuesday in northern Gaza which he described as unprovoked.
The Palestinian Ministry of Health identified the deceased as 34-year-old Ali Ghandour, 23-year-old Inas Khamash, and her 18-month-old daughter, Bayan Khamash.
The Israeli military said seven people were wounded in southern Israel.
The Gaza clashes reached their peak on May 14, marking the 70th anniversary of the Nakba Day (the Day of Catastrophe), which this year coincided with the USA embassy relocation from Tel Aviv to occupied Jerusalem al-Quds.
United Nations envoy Nickolay Mladenov called on both Israel and Hamas to "step back from the brink" and contain the current escalation before it deteriorates with "devastating consequences for all people".
Protests and clashes along the Gaza border since the end of March have seen at least 165 Palestinians killed by Israeli fire.
In exchange for any truce, Israel wants to recover the bodies of two soldiers killed in Gaza and gain freedom for two of its civilians who wandered into the enclave.