Israeli forces battling Palestinian fighters west of Beit Lahiya in the northern Gaza Strip on Thursday afternoon killed six militants, according to Palestinian medical officials.
They said two people were killed in a gun battle, and witnesses said four others were killed in an air strike.
The army had no information on the shooting, but said it had carried out an air strike against a group of gunmen.
An earlier air strike near the southern town of Khan Younis killed two Palestinian civilians, witnesses said. At least nine were wounded.
An Israeli army spokeswoman said aircraft had attacked militants who fired seven anti-tank missiles at Israeli forces. She said the two dead were militants.
One civilian was also killed after being hit as an Israeli tank responded to an anti-tank rocket fired by gunmen, his family said.
A Hamas fighter was also killed in an air strike.
Meanwhile the Palestinian militant group, the Popular resistance Committees, claimed to have shot dead an Israeli soldier during the clashes in Beit Lahiya.
Israeli forces stepped their offensive against Palestinian fighters in Gaza on Thursday.
Ehud Olmert, the prime minister, ordered tanks to push deeper into Gaza overnight after gunmen from the ruling Hamas movement fired rockets into an Israeli city.
Tanks moved inside the western part of Beit Lahiya, where they confronted fighters from Hamas, Islamic Jihad and also the Al-Aqsa Martyrs Brigade of Mahmoud Abbas's Fatah movement, according to witnesses.
The tanks and helicopter gunships fired at militant positions, the witnesses said. The army denied that tanks were inside Beit Lahiya and said ground troops were doing most of the fighting.
The Israeli defence minister, Amir Peretz, said that although Israel withdrew from Gaza last year, "no one should see that as a guarantee that we cannot reach territory in which we feel we have no choice but to operate".
"We have no intention of sinking into the Gaza swamp," he said.
Tanks earlier pushed into the ruins of three of the 21 Jewish settlements evacuated when Israel left Gaza last year.
The military said Israeli forces would stay in the area, several kilometres inside northern Gaza, "until the completion of their mission". Rockets are often fired from the former settlements at Israel.