Witnesses said a Hamas policeman was killed in an earlier clash at the al-Shifa hospital in Gaza City.
Nour Odeh, Al Jazeera's Gaza correspondent, said: "No measures have been made to protect ordinary civilians on the ground. Clashes have become uglier, and exchanges have even taken place next to a hospital".
Rival Palestinians have also fought outside the headquarters of the Fatah-controlled general intelligence agency in Gaza, a Palestinian security source said.
The source said the shooting started when Hamas men fired a rocket-propelled grenade at the agency's compound in northern Gaza.
|School girls run past a Hamas fighter [AFP]|
The person killed in the fighting at the hospital was reported to be Ismail Abu al-Kheir. At least five children are also among the wounded.
Sofiane Abu Zaida, a senior Fatah official, was freed two hours after his car had been stopped by members of the Ezzedine al-Qassam Brigades, the armed wing of Hamas.
"We are fighting each other, about nothing," he said.
A total of 13 members of Hamas and Fatah were kidnapped in various incidents in the Gaza Strip, violating the ceasefire agreed upon a day earlier.
Fatah sources told the Reuters news agency that two abducted men had been "executed" by a Hamas-led police unit. A Hamas police spokesman denied anyone had been abducted or killed.
Clashes between Hamas and Fatah loyalists broke out after the Palestinian president and the head of Fatah, called at the weekend for early elections.
Hamas has rejected the call and accused Abbas of attempting to stage a coup.
On Sunday both sides agreed to a ceasefire that stipulated, among other things, an end to all confrontation and the withdrawal of guns and fighters from the street.
The pact also called for an end to incitement through the media, such as radio, and a stop to all demonstrations and rallies.
King Abdullah II of Jordan has offered to host talks between Abbas and Ismail Haniya, the Palestinian prime minister, to resolve the situation.
"Jordan is willing to do all it can to help the Palestinians overcome their differences and to bolster Palestinian unity," a statement from the royal palace said.
Hamas said it had not yet received the Jordanian invitation.
"If we receive a formal invitation, it will be studied and addressed in harmony with national interest," Fawzi Barhoum, the group's spokesman, said. "We are with all for calls for unity and national interest, but not at the expense of Hamas."
Ahmed Abul Gheit, the Egyptian foreign minister, has urged Palestinian factions to abide by their truce.
Abul Gheit warned against the "consequences of spiralling internecine fighting between the Palestinian factions" and asked Hamas and Fatah "to abide by the cessation of hostilities" agreed upon on Sunday.
Source: Al Jazeera and agencies