Reports from Gaza say fierce fighting has broken out between Israeli troops and Palestinian gunmen near the Kissufim border crossing.
The Israeli army says one soldier was killed and another three injured when a bomb exploded beside their jeep.
In northern Gaza, a member of a militant group had been killed in an Israeli air strike, medics said.
The clashes come three days after Israel ended a major offensive in which more than 120 Palestinians were killed.
The Israelis say the incursion was aimed at stopping rocket fire into Israel by Palestinian militants.
Two Israeli soldiers and an Israeli civilian were also killed over the same period.
Israeli tanks were rushed to the area in which the Israeli jeep was attacked and helicopters landed to evacuate the wounded, eyewitnesses say.
One of the injured was seriously wounded, the Israeli military said.
An official said the jeep was on a patrol in Gaza near the fence that marks the border between the Israel and the strip.
Also on Thursday, a coalition of UK-based human rights and development groups published a report saying Gaza's humanitarian situation is at its worst since Israel occupied the territory in 1967.
The report criticises Israel's blockade on Gaza as illegal collective punishment which fails to deliver security.
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