Violence erupted in Gaza, where Israeli fire killed a Palestinian and militants fired the first rocket into Israel's Beersheva since a 2009 offensive.
One Palestinian fighter was killed and 10 other people wounded when an Israeli tank opened fire in clashes with Palestinian militants east of Gaza City, Palestinian officials said.
"Two of them are in critical condition," Adham Abu Selmiya of the Hamas-run rescue services told AFP after the wounded were taken for treatment to the city's Shifa hospital.
Three of them were militants from Al-Quds Brigades, the armed wing of the radical Islamist Jihad movement and one of them later died, the group said.
It said its fighters were wounded while "firing two mortar shells towards Israeli tanks which were operating inside the (Gaza) border".
The Israeli army said troops opened fire after coming under attack.
"This morning an Israel army force was engaged in a regular patrol along the Gaza border when an explosive device was detonated near the force and a mortar shell fired at it," a military spokeswoman said.
"The force spotted several terrorists and opened fire on them, identifying a hit," she said, adding that militants had placed 12 bombs along the border fence in the past two months.
Palestinians later fired a rocket at the Israeli town of Beersheva, home to about 200,000 people 40 kilometres (25 miles) from the Gaza Strip, but there were no injuries, reported Israeli army radio.
It was the first Grad rocket attack on Beersheva since Israel's devastating 2009 offensive on the Gaza Strip designed to end such attacks.
Israeli television broadcast footage of a crater gouged out of a street next to a badly damaged house.
Local residents ran to shelters after warning sirens went off before the impact, television reported, while emergency services raised the alert level in southern Israel, braced for further attacks.
Al-Quds Brigades said earlier it fired three mortars into Israel, targeting Kibbutz Nahal Oz, east of Gaza City. The military confirmed mortars had fallen in the region, without causing casualties.
Israel public radio said one of the mortars hit a football field.
In a separate incident, another two Palestinians were wounded by Israeli gunfire as they collected gravel north of the town of Beit Lahiya near the border, he said.
And in Khan Yunes, southern Gaza, a seven-year-old child was killed and two other people were wounded in an explosion apparently set off by accident and which also badly damaged a home, witnesses said.
