One Palestinian was killed in a new Israeli air strike in Gaza, hours after Egypt apparently brokered a ceasefire.
Another person was injured in the attack, in the southern border city of Rafah. Israel said the pair were preparing to fire a rocket.
Egypt had sought to broker a truce after retaliatory attacks killed nine Palestinian militants in Gaza and an Israeli civilian in Ashkelon.
The violence is the most serious since a prisoner exchange between the sides.
Earlier this month, 477 Palestinian prisoners were released from Israeli jails in a swap for the captured Israeli soldier Gilad Shalit.
A further 500 Palestinian prisoners are due to be freed later this year as part of the deal between Israel and Hamas, the Islamist militant group that governs in Gaza.
BBC Gaza correspondent Jon Donnison says an escalation of violence could jeopardise those releases.
Amongst the first batch of releases were members of Islamic Jihad, the group involved in the violence on Saturday.
In the latest attack, witnesses named the dead man as Ahmed Jarghun, a militant with the armed wing of the Democratic Front for the Liberation of Palestine (DFLP), according to the AFP news agency, which said the injured man was also a DFLP militant.
The ceasefire had reportedly been due to start at first 01:00 GMT or 04:00 GMT.
Israeli officials said the exchange of fire was continuing into Sunday.
Police spokesman Micky Rosenfeld told AFP news agency that 12 rockets were fired into Israel overnight, the last shortly before 07:00 local time (05:00 GMT).
According to the AP news agency quoting an Israeli military statement, Israeli aircraft targeted six militant sites in Gaza overnight.
Israeli officials have closed schools in southern areas within 25 miles (40km) of Gaza, as well as a university in Beersheba and several colleges, which were to have begun their academic year on Sunday.
After the ceasefire times passed, a senior Islamic Jihad official told AFP that his group was still committed to a truce as long as Israel "commits to it".
Israel said at 38 rockets and mortars were fired into southern Israel since Saturday, killing one man in the coastal city of Ashkelon. Four other people were wounded by the attacks.
A Gaza Health Ministry spokesman said nine people had been killed and 15 wounded in separate attacks on militant targets.
An Israeli military spokesman confirmed a total of four strikes in Gaza, saying the military hit Islamic Jihad militants.
The spokesman said that the first attack, about midday local time, specifically targeted a cell responsible for a long-range rocket attack on Wednesday that exploded deep inside Israel. That attack had caused no casualties.
Israel's military said its aircraft "targeted a terrorist squad... that was preparing to launch long-range rockets".
Islamic Jihad confirmed that five militants had been killed in the attack on Rafah, including one of its commanders, Ahmed al-Sheikh Khalil. Four more militants had been killed in the later air strikes.
A spokesman said Khalil was one of the group's chief bomb makers.
"Today it was a great loss for us in the Islamic Jihad," he said. "The size of our retaliation will equal our loss."
Islamic Jihad is one of the main militant groups in Gaza but is not directly affiliated with Hamas.
The group released photos of rockets being launched from the backs of pickup trucks. It said this was the first time they had used this system; in the past it has fired them from ground launchers.