Hamas later claimed responsibility for firing five rockets from Gaza into Israel, one of which landed on a house in the town of Sderot.
One person was seriously injured in the rocket attack, rescue workers said.
It was the first Hamas rocket attack on Israel in three weeks. Hamas members have previously said that they would observe a ceasefire with Israel.
Colonel Ali Qayssi, of Fatah, said that dozens of members of a Hamas paramilitary force and the faction's military wing, the Izz al-Din al-Qassam Brigades, fired grenades and anti-tank fire on the military base near Karni, where 200 guardsmen are stationed.
When other Fatah members arrived to drive off the attacking force, they were ambushed by other Hamas fighters, he said.
Fatah men who survived the ambush said that the Hamas men had killed any Fatah men who they had captured.
An unidentified Fatah security man who said he had been wounded in the incident told Palestinian television: "They came and shot the wounded. They left me, believing I was dead."
A spokesman for the Hamas paramilitary, the Executive Force, which is controlled by the interior ministry, denied its members were involved and blamed Israel.
Earlier on Tuesday, a fighter from the Izz al-Din al-Qassam Brigades was shot dead and another Hamas supporter wounded in Gaza City, medics and witnesses said.
Ibrahim Mounia was killed in eastern Gaza City overnight when unknown armed men ambushed a minibus carrying men from the brigade, sources said.
Another Hamas loyalist was wounded when fighters opened fire near a checkpoint controlled by national security, a service dominated by Fatah loyalists.
"The new government should show signs of maturity and responsibility"
Husky, Ottawa, Canada
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At least 23 Palestinians have been killed in factional fighting since Sunday and more than 60 wounded, dealing a major blow to the unity government just two months after it was formed.
Another blow to the administration was al-Qawasmi's resignation which has left Ismail Haniya, the prime minister, assuming responsibility for the interior ministry until a replacement is found.
In a bid to clamp down on the street fighting, Haniya is expected to convene a meeting of commanders on Tuesday to discuss a prospective security deployment.