Al Jazeera's Nour Odeh in Gaza said that while most members of the Hellis clan are loyal to Fatah, led by Mahmoud Abbas, the Palestinian president, they insist that the clashes are a family matter, and are not related to their political affiliations.
Odeh said: "Mediations efforts have been going on all day today and they have yet to succeed in bringing this very fierce gunbattle to an end."
On Wednesday, street battles had erupted between the Hellis clan and the Executive Force, a paramilitary unit that has policed Gaza since the group seized control in June.
Four people were killed in that round of fighting and another 30 were wounded, making it the deadliest bout of infighting since Hamas took control of Gaza four months ago.
The next day, the interior ministry and the clan agreed to a ceasefire, in which the gunmen would withdraw and the police would leave the area.
However, the ministry accused the clan of breaking the agreement.
Ihad al-Ghussein , a spokesman for the ministry, said: "Since last night the members of the Hellis family have shown they have no intention of implementing our agreement and have erected sand blockades, laid mines and taken up arms."
He also said that a fighter from the clan had opened fire on members of the Ezzedine al-Qassam Brigades, the armed wing of Hamas, wounding one fighter.
"Another tried to fire an RPG [rocket-propelled grenade] at Qassam members but it exploded in his hands and killed him instantly. At that moment the clashes broke out."