Palestinian militants demanding reform of the security services have clashed for a second night with intelligence staff in the southern Gaza Strip.
Musa Arafat is seen by militants as a byword for cronyism
The violence began after Palestinian leader Yasser Arafat named one of his close relatives as security chief.
Correspondents say the crisis poses a serious challenge for Mr Arafat.
The Palestinian cabinet is to meet on Monday and Prime Minister Ahmed Qurei is expected to announce whether or not he will carry out a threat to resign.
Mr Qurei tendered his resignation on Saturday after a string of high-profile kidnappings - but his offer was rejected by Mr Arafat.
At least 12 people were wounded when members of the al-Aqsa Martyrs' Brigades exchanged fire with officers in the town of Rafah late on Sunday.
The militants have taken violent exception to the appointment of Musa Arafat on Saturday.
They view it as a sign that cronyism is rife within the Palestinian Authority, and demand that he step down.
Musa Arafat, for his part, said he was prepared to fight all "potential enemies" and would ignore the protests.
"I take my orders from His Excellency President Arafat," he said.
"The one who appointed me is the only one who can ask me to quit my job."
After Musa Arafat took up his post on Sunday, supporters of the al-Aqsa Martyrs' Brigades marched in Rafah, on the southern edge of the Gaza Strip.
A fierce gun battle erupted as gunmen attempted to storm the headquarters of military intelligence in the city.
Armed supported of the al-Aqsa Martyr's Brigades - an offshoot of Yasser Arafat's own Fatah movement - marched in other Gaza Strip cities on Sunday.
There were chants of "No to Musa Arafat, yes to reform",
at a rally at the Nusseirat refugee camp in central Gaza.
Earlier in Khan Yunis, al-Aqsa gunmen burned down a post manned by security personnel.
The group has said it does not accept the nomination of Musa Arafat and will continue to take "tough action" until he steps down.
The dispute over the appointment comes against a background of pressure from militants for extensive reform of the Palestinian Authority, which they say is corrupt.
Mr Qurei tendered his resignation after a string of high-profile kidnappings on Friday.
The al-Aqsa Brigades are turning against their own government
Among those abducted and then released were the Gaza police chief, Ghazi Jabali, and four French aid workers.
Mr Qurei described the security situation in Gaza as "a real disaster, a real catastrophe, and an unprecedented lawlessness".
Mr Arafat rejected his offer to stand down. And, in a separate move, he announced an overhaul of security, cutting the number of services from eight to three - a long-standing demand of international mediators.
But, after four hours of talks between the two men on Sunday, Mr Qurei insisted he still intended to go.
The BBC's David Chazan in Jerusalem says many Palestinian officials fear they may now be facing a period of prolonged political instability.