Rival clans clash in the southern town of Khan Younis.
In two other incidents, unknown assailants fired shots at the house and car of a senior security officer who has had past problems with Hamas.
Shots were also fired at a police headquarters, and police returned fire, the security officials said.
The fighting came as negotiators from Hamas and Fatah tried to resolve lingering issues over a national unity government.
Hamas, which won parliamentary elections last year, had agreed last month to form the coalition government with Mahmoud Abbas, the Palestinian president and leader of the Fatah party, partly in hopes of ending months of deadly street battles between the two groups.
Talks not done
“The Palestinian people are in desperate need of leaders to solve their daily problems and security”
Ismail Haniyeh, the Palestinian prime minister and head of Hamas, said on Monday that talks on the government were not done.
“The Palestinian government won’t finish discussions or be announced before the end of next week,” he said.
Haniyeh has two more weeks to form a government.
Meanwhile, security officials said they believed Muslim extermists were behind the bombing of a music shop in the southern town of Khan Younis early on Monday, following a similar attack on an Internet cafe on Sunday.
Another Internet cafe owner was also briefly kidnapped. Nobody was injured in the attacks.