Israel's military began targeting militant sites in Gaza on Wednesday, calling the attacks a response to earlier rockets that were launched from Gaza, one of which struck a residential home in southern Israel.
Israel Defence Minister Avigdor Lieberman ordered the closure of both border crossings with the enclave.
Another rocket from Gaza struck a beach near one of Israel's larger cities.
In this image made from video, police inspect the damage to a building from a rocket, Wednesday, Oct. 17, 2018, in Beersheba, Israel.
"At the same time, we emphasize we are ready to confront the Israeli aggression".
"At 4am [01:00 GMT] Israelis in the city of Beer Sheva were running to bomb shelters after a rocket was launched from the Gaza Strip at Israel", the Israeli army said on Twitter.
Beersheba, which is some 30 miles from Gaza, is the biggest city in southern Israel, with a population of around 200,000.
Israel struck almost two dozen Hamas military targets throughout Gaza, Conricus added.
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Reports Wednesday said that Abbas Kamel, head of the Egyptian General Intelligence Services, had cancelled a trip to Gaza, the West Bank and Israel set for Thursday amid the tensions surrounding Gaza.
At least three other Palestinians were wounded in the strikes, the Gaza health ministry said.
It comes after months of violent Palestinian protests on the Gaza border, sparking deadly gunfire from Israeli troops and fears of an all-out conflict between Hamas and Israel, which have fought three wars since 2008.
Hamas has held weekly border protests for the past six months, aimed in part at easing a crippling Israeli-Egyptian blockade that was put in place after the Islamic militant group violently seized power in 2007.
The rioters carried burning tires, Molotov cocktails, hand grenades, rocks and live fireworks, all of which they threw at Israeli soldiers who were stationed on the other side of the border, to secure the region from terrorist infiltration. Seven Palestinians were killed, including four who were shot while approaching a military position inside Israeli territory.
The situation in southern Israel further deteriorated last Friday when an estimated 15,000 Palestinian Arabs participated in an orgy of violence along the border with Israel, while Hamas used the chaos to launch another attempt to infiltrate the Jewish state and to kidnap IDF soldiers. Over the weekend, Israel halted Qatari-donated fuel shipments to Gaza's power plant in response to escalating violence. Prime Minister Benjamin Netanyahu threatened "very painful blows", saying Israel was very close to waging a "different kind of activity".
Hamas has been ramping up the violent riots along the border fence in order to achieve the complete removal of what it calls the Israeli "blockade" or "siege". Egypt also restricts movement in and out of Gaza on its border.