The Izzedine Al Qassam Brigades said its fighters shelled the Zikim military base near Ashkelon and the Israeli cities of Ashdod and Beer Sheva.
The group also claimed Saturday to have shot down an Israeli F-16 over the Mediterranean Sea. It didn't say when the downing allegedly occurred.
An IDF spokeswoman denied the claim, saying all aircraft are accounted for.
"It didn't happen and we have no record of it," said the spokeswoman, who could not be named in line with policy.
Hamas accused Israeli warplanes of hitting an apartment building Saturday in the Jabalya refugee camp in northern Gaza, killing at least three people and wounding more than 30, according to Hamas-run al Aqsa TV. Footage broadcast by al-Aqsa showed people clamoring over smoking ruins, searching for survivors.
Palestinians incensed by Israel's offensive took to the streets in the West Bank, the Palestinian territory whose government is dominated by the Fatah movement.
Four people were seriously wounded in Beitanya near Ramallah when Israeli soldiers fired "high-velocity projectiles" toward demonstrators, Palestinian medical sources said. Seven others were lightly injured by rubber steel-coated bullets and tear gas inhalation, the sources said.
In Bait Omar, demonstrators attempted to march out of the village but were stopped by Israeli soldiers, who fired stun grenades and tear gas.
Fatah and Hamas have been fierce rivals in recent years. Fatah is more moderate on Israel than Hamas, and Hamas leaders have engaged in peace talks with Israel over the years.
Q&A: What is Hamas?

Comments