In another attack, anti-tank rockets and mortar shells were fired at Fatah's police headquarters.
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
At least eight people were wounded in exchanges of fire between the sides overnight, Palestinian security officials said on Sunday.
Major roads in Gaza City were blocked by concrete barriers put up by security personnel loyal to both factions, causing traffic jams as drivers were forced on to alternate roads.
Large security details loyal to both groups deployed at major street corners and outside potential targets, like the homes of Mahmoud Abbas, the president and leader of secular Fatah, and Ismail Haniya, the prime minister from the Islamist Hamas movement.
Security was also reinforced around Palestinian media outlets, and two Gaza City universities were shuttered.
Late on Saturday night, a 12-year-old boy was killed and two people were wounded in northern Gaza during a shootout, witnesses and hospital officials said.
The new death brought the Palestinian death toll from the fighting in the last three days to more than 22.
The boy's father, Baher Abu al-Karaya, a local Fatah leader, was injured in the gun battle. From his hospital bed, he told The Associated Press that Hamas fighters attacked his home.
Plea for calm
Hamas and an Abbas spokesman appealed for calm on Saturday.
But referring to unity talks, Ayman Taha, a Hamas official, said on Saturday: "As long as the leaders of the coup within Fatah are continuing the killings, there can be no dialogue."
For his part, Tawfiq Abu Khoussa, a Fatah spokesman, blamed Hamas for starting the latest round of fighting.
"How can dialogue go on when there is a bomb underneath the table?"
"We in Fatah did not stop the dialogue," he said, referring to talks on a national unity government.
The streets of Gaza City were empty on Saturday after the previous day's fighting and shops and markets remained closed.
But Fatah and Hamas clashed again near the Islamic University, killing at least four in some of the fiercest fighting since Hamas's election victory a year ago.
One of the dead was believed to be a university student who was caught in the crossfire, witnesses said. Several people were also injured in the clash.
A member of the Hamas-led police force died of his wounds overnight, hospital officials said, bringing the toll from Friday's factional fighting to 18.
In the latest of a string of tit-for-tat abductions, four members of a security force loyal to Fatah, including Abed Abdeen, a local commander in the southern Gaza Strip, were taken captive by unknown fighters.
Abdeen's relatives later said that seven Hamas men in the southern town of Khan Younis had been seized in retaliation.
Hamas mounted an overnight raid on the Gaza headquarters of the Preventive Security Service, a force dominated by Fatah members, who are loyal to Abbas.
|A Palestinian girl walks through destroyed|
buildings in the northern Gaza Strip [EPA]
Hamas attacked the compound with mortars and rocket-propelled grenades but no injuries were reported.
Grenades were also fired at the home of Rashid Abu Shbak. It was unclear whether Shbak was home at the time, but no injuries were reported.
Fighting also took place outside the homes of Abbas and Mahmoud al-Zahar, the Hamas foreign minister. Hamas officials said al-Zahar's home was damaged by two rocket-propelled grenades.
Hamas has said any new elections would amount to a coup.
Abbas, currently in Madrid, has said he will push ahead with his plan to call early elections if coalition negotiations fail within two to three weeks.