A continued ceasefire in Gaza is of "critical importance", Barack Obama's Middle East peace envoy, George Mitchell said today, as Israeli jets bombed smuggling tunnels under the Gaza-Egypt border amid the worst violence in the territory since a truce began 10 days ago.

Mitchell arrived in Israel this afternoon after talks in Cairo with the Egyptian president, Hosni Mubarak. In Jerusalem, he held talks with the Israeli president, Shimon Peres, and was due to meet the prime minister, Ehud Olmert, before travelling to the West Bank to see the Palestinian president, Mahmoud Abbas.

"It is of critical importance that the ceasefire be extended and consolidated, and we support Egypt's continuing efforts in that regard," Mitchell told reporters after talking with Mubarak. "The United States is committed to vigorously pursuing lasting peace and stability in the region."

The former Northern Ireland peace broker's mission – which he described today as "clear and tangible evidence" of Obama's commitment to the Middle East – comes with tensions in Gaza as high as they have been since the end of the Israeli attack on the territory, in which about 1,300 Palestinians died.

People living in Rafah, by the Egyptian border, fled their homes as Israeli aircraft struck three times before dawn. There was no immediate news of any casualties.

The attack on the tunnels, used to smuggle weapons and other goods, took place a day after an Israeli soldier was killed by a roadside bomb while patrolling the border between Gaza and Israel. Three other soldiers were injured.

Israel launched a retaliatory airstrike which killed a Palestinian man travelling on a motorcycle. Israel said he was the planner of the roadside bombing, while Hamas, which controls Gaza, said only that he was a Hamas member. The West Bank is run by Abbas's Fatah movement, which was expelled from Gaza by Hamas in mid-2007.

The heightened tensions were illustrated today as French officials summoned Israel's ambassador to Paris to protest at an incident in Gaza in which Israeli troops fired warning shots at a car containing a French diplomat.

France's foreign ministry said the vehicle, containing the French consul general and other European diplomats, was held at the Erez crossing between Gaza and Israel yesterday for six hours, during which time Israeli troops fired two warning shots, the AFP news agency reported.

Obama signalled the seriousness of his intention to engage in the search for a Middle East peace agreement by giving his first foreign interview to the al-Arabiya television channel on Monday.

"Sending George Mitchell to the Middle East is fulfilling my campaign promise that we're not going to wait until the end of my administration to deal with Palestinian and Israeli peace. We're going to start now," Obama told al-Arabiya, which is based in Dubai but is broadcast to most of the Middle East.

"He's going to be speaking to all the major parties involved. And he will then report back to me. From there we will formulate a specific response," Obama said.

The Obama administration appears intent on trying to help the Palestinians while being seen not to abandon its traditional support for Israel. The new US secretary of state, Hillary Clinton, reassuring Israelis, today backed Israel's bombardment of Gaza.

"We support Israel's right to self-defence. The [Palestinian] rocket barrages which are getting closer and closer to populated areas [in Israel] cannot go unanswered," Clinton said in her first news conference at the state department.

She added: "It is regrettable that the Hamas leadership apparently believes that it is in their interest to provoke the right of self-defence instead of building a better future for the people of Gaza."

Hamas has not claimed responsibility for yesterday's bombing, but described it as "a natural response" to Israeli policies.

The Israeli military said it saw Hamas as "accountable for preserving the peace in Israel's southern villages and will respond harshly to any attempt of undermining it".