GAZA CITY (Agencies) - Israel began withdrawing troops from the Gaza Strip on Sunday after the Islamist movement reciprocated to a ceasefire which was declared by the Jewish state on Saturday night after its deadliest 22-day offensive ever launched on the battered Palestinian territory. After exchanges of gunfire and an airstrike punctured what Israeli Prime Minister Ehud Olmert acknowledged was a "fragile" unilateral ceasefire, Gaza's Hamas rulers and other armed groups announced their own one-week ceasefire to allow Israeli troops to withdraw from their territory. "We in the Palestinian resistance movements announce a ceasefire in the Gaza Strip and demand that enemy forces withdraw in a week and open all the border crossings to permit the entry of humanitarian aid and basic goods," Mussa Abu Marzuk, the deputy leader of Hamas' politburo, said in Damascus, as medics scrambled to pull dozens of bodies from the mountains of rubble left by Israel's three-week offensive. Dawud Shihab, a Gaza-based spokesman for Islamic Jihad, a smaller armed faction, said the truce would give an opportunity for Arab governments to put pressure on Israel to withdraw all its troops after the three-week blitz. "During this period, the resistance is ready to respond to all efforts by the Egyptians, Turks, Syrians and Arabs," he told AFP. Another Hamas spokesman warned that the group would not accept the presence of a single Israeli soldier in Gaza. "We have clearly said: if Israeli troops remain in Gaza, this will be a wide window for the resistance against the occupation," Osama Hamdan, the group's representative in Lebanon, said in an interview with Al-Jazeera television. Hamas activists in the Gaza Strip, however, fired at least 18 rockets and mortar rounds at Israel following the announcement, wounding at least one person, the Israeli Army claimed. The Army confirmed troop withdrawals had begun after witnesses saw Israeli tanks and troops pull back from key positions in and around Gaza City towards the border fence which surrounds the impoverished territory. "I can confirm that there is a gradual withdrawal from the Gaza Strip," an Army spokeswoman told AFP. Israeli Prime Minister Ehud Olmert said on Sunday he wants to withdraw troops from "as quickly as possible" after having declared a unilateral ceasefire. "We are not interested in staying in the Gaza Strip, we want to leave as quickly as possible," Olmert told a gathering of European leaders who arrived in Jerusalem after attending a Gaza war summit in Egypt. After the ceasefire came into effect at 2:00 am (0000 GMT), Gaza enjoyed its first bomb-free night in more than three weeks, but there were soon signs that the calm may unravel. As Hamas men fired rockets and Israel launched retaliatory airstrikes, troops shot dead an eight-year-old girl in the northern town of Beit Hanun and a 20-year-old man near Khan Yunis in the south, medics said. Gaza medics took advantage of the halt in the fighting to rush to areas which had been inaccessible, pulling at least 95 bodies from the rubble, including those of several children. The discoveries brought the overall death toll since Israel launched Operation Cast Lead on December 27 to at least 1,300, medics said, making it by far the deadliest Israeli offensive ever launched on the Gaza Strip. On the Israeli side three civilians and 10 soldiers were killed in combat and rocket attacks. On the ground, as Hamas congratulated the Palestinians on "victory" from mosque loudspeakers, Gaza residents cautiously ventured out onto the streets, surveying the rubble that was once their homes. Palestinian President Mahmud Abbas repeated his call for a complete Israeli withdrawal from Gaza and the reopening of the enclave's border crossings, saying Israel's truce was "important and necessary but insufficient." Meanwhile, Israel's intelligence chief admitted that the offensive failed to wipe out Hamas' "network of arms smuggling tunnels" below the border with Egypt which Palestinians will now likely rebuild. Speaking at the end of the weekly cabinet meeting, Yuval Diskin said he feared the situation along the border, known as the Philadelphi route, would return to the pre-war status quo unless agreements with the US and Egypt on greater surveillance are followed through. "Not all of the tunnels have been destroyed. The second calm is restored and if Israel does not insist on the implementation of agreements on the issue, the situation along the Philadelphi route will return to its previous state within several months."