Machinegun fire could be heard outside the camp on Sunday as the army continued its siege backed by helicopter gunships and tanks.
Explosions rocked the camp but witnesses said the fighting was less intense than the previous two days.
Zeina Khodr, Al Jazeera's correspondent outside the camp, said Fatah al-Islam confirmed that Abu Riyadh, one of its leaders, was killed by an army sniper on Saturday.
Khodr said there had been conflicting reports in previous days that other senior Fatah al-Islam leaders had been killed, reports that were denied by the group.
She said the army had managed to reach a few hundred metres inside the camp and clear out pockets of fighters.
More than 16 people, fighters and civilians, have died in the camp over the weekend. Fatah al-Islam said it has lost three fighters.
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The Lebanese government has said it aims to eliminate Fatah al-Islam once and for all.
Fouad Siniora, the Lebanese prime minister, said on Saturday that the only option for members of the group was to give up their weapons and surrender.
"This is a terrorist gang," he said in a TV interview. "They have to surrender themselves and their arms."
But Abu Salim Taha, a Fatah al-Islam spokesman inside the camp, said: "There is no way we will give up our weapons because it is our pride. We cannot even contemplate surrendering."
Military helicopters fired two rockets and machinegun barrages at targets on the camp's coastal side.
"Since yesterday morning, the shelling has been ongoing all over the camp. Two shells fell on the building I'm in now. Several buildings have collapsed," a Palestinian resident inside the camp said.
"There's only one clinic with one doctor left. There's no electricity, bread or medicine."
Jihad Omar, who is working with rescue teams in the centre of the camp, told Al Jazeera on Saturday that casualties were mounting and that wounded people were lying in the streets.
|Lebanese soldiers stake out the Nahr al-Bared|
camp as part of a continuing siege [AFP]
He said shelling continued in the centre of the camp where there were no Fatah al-Islam fighters, only civilians.
The army has urged refugees still trapped inside the camp to "be patient and to expel those criminals from among you".
Siniora said that the camp's population had fallen from more than 31,000 to fewer than 3,000, including the fighters, after civilians fled after the outbreak of fighting on May 20.
He accused Fatah al-Islam of preventing the remaining civilians from leaving the camp in an apparent bid to use them as human shields.