The pilot was believed to be the only person on board the Cessna, which crashed down on Israel's Mediterranean coast, about 30 miles north of Tel Aviv, the army added.
The plane took off from the Lebanese capital Beirut, and was spotted by the military as it crossed into Israel, the army said.
Israeli warplanes and helicopters made repeated attempts to contact the Cessna, but the pilot refused to respond or identify himself. After 15 minutes, the plane was shot down, said army spokesman Jacob Dallal.
Most of the plane's wreckage crashed down on a navy training school on the beach near the resort town of Mikhmoret, 30 miles north of Tel Aviv. The pilot's mangled body fell through the roof and into the office of the school's assistant principal, Nitza Kramer, who was treated for shock.
"Suddenly from the corner of the ceiling a flash of light came through and then everything went black and the room was full of smoke," Kramer told The Associated Press. "I couldn't see anything but I stepped on something soft which must have been the body."
The Israeli army was on alert for possible attacks by the Lebanese guerrilla group Hezbollah on Thursday, the first anniversary of Israel's military withdrawal from south Lebanon.
We took into account that in light of the threats by Hezbollah, this was an attempt at a terrorist attack on an Israeli city, said army spokesman Lt. Col. Olivier Rafowicz.
He said the army had been fearing a suicide plane attack.
The Israeli aircraft fired warning shots, and there also was eye contact and signs made at (the pilot), Rafowicz said.
There was no immediate word on whether the plane was carrying any weapons or explosives.
In Lebanon, the pilot was identified as Stefan Nikolian, 43, a Lebanese of Armenian heritage, said Nemeh Malek, director of the plane's owner, JR Executives.
Malek told The Associated Press in Beirut that the pilot had flown three or four times this week with a trained pilot.
Today, he got on the plane, started the engine and took off, leaving the pilot behind, Malek added.
Nikolian was not licensed to fly without a trainer, Malek said.
Asked what prompted the pilot's actions, Malek replied: Nobody knows.
As the plane taxied, its wing hit an airport worker, slightly injuring him, security officials said.
The Lebanese army said Nikolian took off without authorization. But the army in a statement said the Israeli aircraft "forced him to head south" and enter Israeli airspace.
Hezbollah, in a broadcast on its radio sation, was first to identify Nikolian. The station did not say whether he was linked to Hezbollah, which fought Israel's forces for years in south Lebanon.
Hezbollah has continued to carry out occasional attacks on a disputed patch of land along the Israeli-Lebanese border.
The Israelis also say that Hezbollah has been assisting the Palestinians with their uprising against the Israelis.
Anti-Israel guerrillas have a history of launching air attacks from Lebanon.
In 1987, a Palestinian guerrilla took off from Lebanon in a hang-glider and landed near a military outpost in northern Israel, killing six soldiers with hand grenades and a machine gun before being killed himself.
In a similar hang-glider incursion in 1981, a Palestinian guerrilla landed in northern Israel and seized a civilian hostage. But he was captured after he fell asleep, allowing his hostage to escape.
Also in 1981, a squad of guerrillas flew a hot-air balloon from south Lebanon into Israel, but were all shot down and killed.
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