“The next few days will be critical and more people may leave the country if the current situation is not resolved peacefully soon,” the statement said.
Foreign troops had entered the country on Thursday as part of a broader international push to persuade Mr. Jammeh to hand over control to Mr. Barrow, who was elected in December.
Mr. Barrow was still in Senegal after having been sworn in on Thursday during a ceremony at the Gambian Embassy in Dakar, the Senegalese capital. It was unclear when he would return to Gambia, but his aides have said he planned to be escorted into the country by Gambian security forces.
Mr. Barrow, a real-estate agent who not long ago was a low-level opposition party member, is among those who have fled Gambia because of security concerns. Mr. Jammeh, who had said he would rule for a billion years, had promised to marshal his troops if an international coalition of forces intervened.
Mr. Jammeh initially conceded the race, before he suddenly called into question the election results. He demanded a new vote and appealed to the Supreme Court. He persuaded Parliament to pass a measure extending his term for three months.
The Supreme Court did not have enough judges to hear his case because Mr. Jammeh had fired most of them and had not replaced them, and the international community considers the parliamentary measure to be invalid.
Mr. Jammeh was under tremendous pressure to leave. The African Union no longer recognized him as president, and the United Nations Security Council supports Mr. Barrow. The international military coalition that entered his country was assembled by the Economic Community of West African States.
A spokesman for Mr. Barrow said that if Mr. Jammeh stepped down, Mr. Barrow could travel to the border to be escorted across by Gambian military forces. Adding to the tension is the looming question of whether the military remains loyal to Mr. Jammeh.
Mr. Jammeh had led Gambia since a coup in 1994. His government was criticized for gross violations of human rights. He jailed opponents and journalists, some of whom died in prison. He talked of decapitating gay people.
An unpredictable president, he once claimed that he could cure AIDS with herbs, prayers and a banana, and he has carried out literal witch hunts, combing the countryside for suspects and forcing them to drink a hallucinogenic potion.
As a result, thousands of people fled long before the recent presidential standoff, with many heading to Senegal, which borders Gambia on three sides. The recent flood of people into Senegal represented a significant portion of the tiny population of Gambia, which has about two million people.
Some of the Gambian diaspora turned out for Mr. Barrow’s inauguration on Thursday, streaming into the sandy streets outside the Gambian Embassy in Dakar. Some, including journalists who say they are too scared to work in their home country, wore T-shirts with #GambiaHasDecided printed on them.
Isatou Dumbuya, owner of a spare parts shop in Gambia, fled to Senegal on Sunday when it became clear that Mr. Jammeh had no immediate plans to step aside.
She was lucky: Her daughter goes to college in Dakar, and she and her children can stay at an apartment in the city until the situation calms. But driving across the border, she said, she saw throngs of people fleeing — “My brothers and sisters,” she said.
Ms. Dumbuya voted for Mr. Barrow, and she showed up outside the embassy during his inauguration hoping that her presence and that of other supporters would demonstrate that he must be accountable once he takes over.
For now, though, she worries about her children missing school and hopes to return soon to a changed Gambia.
“When we go back home, we go back home as heroes,” she said.