Israeli Prime Minister Ehud Olmert has ordered the military to continue its offensive in Gaza and urged patience in the quest to free a captured soldier.
Israeli forces have pounded Gaza by air and land
Speaking after the expiry of a militants' deadline to free Palestinian prisoners, Mr Olmert said kidnappings would not "bring Israel to its knees".
One of the groups holding the soldier said earlier they would not kill him.
On Tuesday a homemade rocket fired from Gaza landed near a school in the heart of the Israeli city of Ashkelon.
No-one was injured in the attack, the deepest a rocket from Gaza has reached inside Israel.
Mr Olmert said the rocket attack marked an "escalation of unprecedented gravity" which would have "far-reaching consequences" for Hamas.
The attack came as Israel pursued a week-old campaign to free Cpl Gilad Shalit, believed to be held in southern Gaza.
Despite the expiry of the militants' deadline, Israel said it believed the 19-year-old tank gunner was alive.
"We know that until now Gilad Shalit is alive, we know that he is injured, that he was seen by a Palestinian doctor a few days ago," government spokesman Avi Pazner told French television.
But Abu al-Muthana, from the Islamic Army - one of three groups holding Cpl Shalit - said there would be no more talks nor information released about the soldier's fate.
Mr Olmert said the fight against Palestinian militants would be drawn out.
"This is a long war. It requires lots of patience, sometimes endless restraint. We have to know when to clench our teeth and to deal a decisive blow.
"I gave the order to continue operations to strike terrorists, those who protect them and those who give them orders," he said.
The Hamas Prime Minister Ismail Haniya, whose Gaza offices were hit in a symbolic Israeli missile strike two days ago, appealed on Tuesday for Cpl Shalit's life to be spared.
"The government... is still calling for preserving the life of the captured Israeli soldier and for him to receive good treatment," Mr Haniya said.
The BBC's Middle East editor Jeremy Bowen says nobody on either side knows how long this crisis will last.
Their public statements are a form of psychological warfare, to show resolution and to put pressure on their opponents, he says.
Hamas' Izzedine al-Qassam Brigades and the Popular Resistance Committees - as well as the Islamic Army - say they are holding Cpl Shalit.
SOLDIER CAPTURE TIMELINE
Sun 25 June: Cpl Shalit captured in cross-border attack
Mon 26 June: Captors propose prisoner exchange
Weds 28 June: Israel enters southern Gaza after air strikes
Thurs 29 June: Israel detains dozens of Hamas officials
Sat 1 July: Captors demand 1,000 prisoners be released
Mon 3 July: Captors issue deadline of 0300 4 July
Tues 4 July: Israel says Shalit still alive
After a deadline to free about 1,500 Palestinian prisoners expired, an Islamic Army spokesman told the BBC "all options" were open to the militants, except killing Cpl Shalit.
He said that was because the captors were Muslims who did not treat their captives as others, such as the US, did.
But he said: "The file is closed - there will be no more negotiations, no more information will be released, there will be no mediation."
International organisations have demanded that militants free Cpl Shalit and they have urged the Israeli military to show restraint.
The Swiss government accused Israel of breaking humanitarian law by inflicting "collective punishment" on the Palestinians.