RAFAH, OCCUPIED GAZA - With fears rising that Israel will move ahead with a long-planned full-scale invasion of Rafah, the United Nations said Friday that more than 100,000 people had fled since Israel ordered people to leave parts of the city and intensified a bombardment that Gazan health officials say has killed dozens of people.

As Israeli troops continued to exchange fire with Palestinian fighters near Rafah on Friday, according to both the Israeli military and Hamas, people were packing up their tents and leaving the southern Gazan city and its surrounding areas where more than a million Palestinians had sought shelter in trucks, cars and donkey carts.

Many of them have already been displaced multiple times by Israel’s war in Gaza over the past seven months.

“Around 110,000 people have now fled Rafah looking for safety,” the main United Nations agency that aids Palestinians, known as UNRWA, posted online on Friday. “But nowhere is safe in the #GazaStrip & living conditions are atrocious.” On Thursday, a U.N. official said that 79,000 people had left since Israel issued its evacuation order.

“The only hope is an immediate Ceasefire,” UNRWA said.

Israel seized control of the Gaza side of the Rafah border crossing with Egypt in what it called a “limited operation,” and intense fighting has continued on the eastern edge of the city since. The Israeli military said on Friday that its aircraft had struck Hamas members and rocket-launching sites at several locations in the Rafah area over the past day, while Hamas said its forces had fired mortars on Israeli troops east of the city.

The Israeli security cabinet agreed on Thursday night to expand the operation in Rafah, two officials said, but it was not clear what that would mean in practical terms.

Fighting continues in other areas of Gaza, and on Friday, the Israeli military said four of its soldiers were killed and two were seriously injured by an explosive device near Gaza City, in the northern part of the territory. Israeli forces seized the north months ago but have been unable to control it completely, repeatedly battling militants there.

In an apparent sign of the militants’ staying power, Hamas took responsibility for a rocket attack, the first one since December that was launched from Gaza and triggered air-raid sirens in the southern Israeli city of Beersheba.

Israel has designated what it calls a safe zone for Gazans fleeing Rafah, including Al-Mawasi, a coastal section of Gaza it has advised people to go to for months. But the United Nations has said it is neither safe nor equipped to receive them.

On Friday, UNICEF’s senior emergency coordinator in the Gaza Strip, Hamish Young, said from Rafah that in his 30 years working on large-scale humanitarian emergencies “I’ve never been involved in a situation as devastating, complex or erratic as this.”

“Yesterday, I walked around Al-Mawasi,” Mr. Young said. “The roads to Mawasi are jammed — many hundreds of trucks, buses, cars and donkey carts loaded with people and possessions.”

“People I speak with tell me they are exhausted, terrified and know life in Al-Mawasi will, again, impossibly, be harder,” he said. “Families lack proper sanitation facilities, drinking water and shelter.”