At least 14 people have died in clashes between rival Palestinian groups Fatah and Hamas, despite the declaration of a new ceasefire.
Palestinian President Mahmoud Abbas has accused elements within Hamas of trying to seize power in Gaza by force.
Three Fatah supporters were killed in a gun battle inside a hospital in the town of Beit Hanoun.
In a statement Mr Abbas - the Fatah leader - said some Hamas figures were "planning a coup against the legitimate [Palestinian] institutions".
In Gaza City, three members of one family, all of them Hamas supporters, were shot dead.
Since Monday at least 16 people have died in fighting between rival groups.
A senior militant from the Fatah-linked al-Aqsa Martyrs Brigades was also killed in the northern Gaza Strip
Earlier, Hamas officials accused Fatah of trying to assassinate PM Ismail Haniya, by firing a rocket-propelled grenade at his home.
Earlier in the day, gunmen fired on a government building where a cabinet meeting was taking place
The building was damaged but no-one was injured.
No-one was hurt in the shooting but PM Ismail Haniya was forced to suspend the meeting and ministers had to move elsewhere, an official said.
This was the third time Mr Haniya had come under fire since Monday.
Over the weekend gunmen fired at Mr Haniya's house and two militants died after being thrown from buildings.
The political and military leaders of Hamas are planning a coup against the legitimate institutions, thinking they will be able to control the Gaza Strip by force Mahmoud Abbas statement
Militant factions then called a truce, but there have been a number of others and they have quickly failed.
Mortars were also fired at the Gaza compound of President Mahmoud Abbas. He was not there at the time.
Mr Haniya's Hamas and the rival militant group, Fatah, are locked in a power struggle that has left more than 50 people dead since mid-May.
In another development, Hamas has declared parts of northern Gaza a military zone and barred Fatah members from the streets.
Weekend violence
Hamas and Fatah on Monday agreed their seventh truce in just over a month to try to end the fighting that has claimed at least 70 lives in recent weeks.
The weekend saw some of the worst fighting in a month of bloodletting.
However, the BBC's Aleem Maqbool says the ceasefire has been followed only by an upsurge in the killings, and in their brutality.
In one incident on Sunday, Hamas militants kidnapped an officer in a Fatah-linked security force and killed him by throwing him from the roof of a 15-floor building.
In one incident, the home a Fatah activist was attacked and three members of his family killed, including a 75-year-old woman and a 13-year-old boy.
The incident sparked fresh clashes throughout Gaza City.
A senior militant from the al-Aqsa Martyrs' Brigades, linked to Fatah, was also killed.
Fighting has claimed at least 60 lives in recent weeksFatah militants attacked the house of a Hamas preacher with rocket-propelled grenades before dragging him away. His body was later delivered to a hospital.
Jamal Abu al-Jedian was shot 41 times in his hospital bed in Beit Hanoun. His brother was also later found dead, Fatah said.
Just before midnight, a Hamas activist was thrown off the 12th floor of a building, security officials said.
"Everybody is shooting at everybody," said one doctor quoted by Reuters.
There were no reports of casualties in the Monday morning attack on the house of Mr Haniya in the Shati refugee camp next to Gaza City.
A Hamas mosque preacher, Mohammed al-Rifati, was shot dead by Fatah gunmen.
It was not confirmed whether he was at home when gunmen opened fire from a nearby block of flats, but his wife, children and grandchildren were, relatives said.
Leaders attacked
It was the first time that Mr Haniya has been an apparent target since renewed factional fighting broke out on 13 May.
On Tuesday morning, Palestinian political leaders became a target.
Hope of escape
Mortars were fired at the compound of Mr Abbas, who leads Fatah, in Gaza City. He was not there at the time.
The latest ceasefire was called to allow pupils to take final school exams.
They crossed all the red lines Hamas spokesman Fawzi Barhoum
Militants pulled back from flashpoint junctions on Monday although shooting still punctuated the morning.
Earlier, gunmen fired a rocket-propelled grenade at the home of his Mr Haniya, a leader of the rival Hamas movement, in the Shati refugee camp on the outskirts of the city.
Matriculation exams are expected to go ahead for tens of thousands of 12th grade students, although many reportedly had to take long detours to avoid gunmen still on the streets.
A spokesman for Hamas said the house had been damaged, but that Mr Haniya and his family had not been harmed.
Daliya Naji, 16, said the fighting had kept her awake all night and could affect her chances of success - which for her could mean leaving Gaza to attend university in Egypt.
Fawzi Barhoum accused Fatah supporters of attempting to assassinate the prime minister.
"I am a good student, but I feel my brain is empty. I can't think anymore and I don't know what to do," she said.
"Hamas has decided to punish the attackers and the killers and it will not be reluctant to punish them without any mercy," Mr Barhoum said.
A ceasefire agreed by Hamas and Fatah in May broke down last week after a member of forces loyal to Fatah was killed.
"They crossed all the red lines."
Hamas - which won legislative elections last year - formed a unity government with Fatah in March, but the deal failed to end their rivalry or the violence.
In two separate incidents on Monday, gunmen fired upon Mr Haniya's house and on a government building where he was holding a cabinet meeting.
No-one was hurt in the shootings but Mr Haniya was forced to suspend the cabinet session and ministers had to move elsewhere, an official said.
Our correspondent says the attacks are an indication of how little respect is being shown to the officials, who have been promising an end to the violence, from the moment it began over a month ago.
Meanwhile, the Israeli military carried out an air strike in northern Gaza, attacking what it said was a site near Beit Hanoun used by Palestinian militants to launch rockets into Israel.
The strike came shortly after militants fired rockets into the southern Israeli town of Sderot, wounding three people.
Islamic Jihad claimed responsibility for launching the rockets.
On Sunday, Israeli Prime Minister Ehud Olmert said his country's military action in Gaza would continue until militants stopped their attacks.
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