The unprecedented violence, the worst since Hamas's shock election win a year ago, has scuttled intermittent talks on forming a national unity government and left the Gaza Strip teetering on the brink of civil war.
Saudi Arabia's King Abdullah bin Abdulaziz Al-Saud offered on Sunday to hold talks in Mecca to stop the "disgraceful" fighting, a proposal welcomed by faction leaders.
"I invite my brothers of the Palestinian people, represented by their leaders ... to a quick meeting in their brotherly homeland Saudi Arabia ... to discuss their differences without any intervention from outside parties," he said in an appeal carried by the state news agency SPA.
Khaled Meshaal, the ruling Hamas movement's political chief, and Mahmud Abbas, the Fatah party leader and Palestinian president, both welcomed the offer, though no date for the meeting has yet been announced.
Several previous efforts to end the feuding, including a rare meeting between Meshaal and Abbas in Damascus, have proved fruitless.
Ismail Haniya, the Palestinian prime minister who belongs to Hamas, on Sunday urged an end to the conflict, which spilled over into the occupied West Bank.
"We call on all the Palestinian people to protect national unity, to make the language of dialogue and reason prevail, to withdraw weapons from the streets, and put an end to the tensions," he said.
But the killings and tit-for-tat abductions have continued in both the Gaza Strip and the occupied West Bank.
Laila El-Haddad interviews members of armed groups allied to Hamas and to Fatah to discover why they do what they do - read An eye for an eye in Gaza
An explosion early on Sunday morning rocked the Gaza City home of a bodyguard of Mohammed Dahlan, a powerful Fatah commander. No casualties were reported.
Anti-tank rockets and mortar shells were also fired at Fatah's police headquarters.
Also on Sunday, Fatah supporters, many of them masked, set ablaze the Nablus offices of the Hamas-controlled education ministry and abducted 11 Hamas members.
Hamas fighters in turn seized Shabn Abu Assar, the chief of the Palestinian security forces for Gaza, but released him after two hours.
Among the victims were an 11-year-old boy caught in the crossfire on Saturday night and a two-year-old child killed by a stray bullet during a firefight in Khan Younis on Friday.
Roadblocks set up
"How can dialogue go on when there is a bomb underneath the table?"
Throughout Gaza, the warring factions have set up roadblocks and continue to trade bursts of machinegun fire, and anti-rocket and mortar shells.
The territory has become deserted, with shopkeepers boarding up shops and staying in the relative safety of their homes.