A worker at The Gambia’s main ferry crossing has told the BBC that thousands of people are fleeing every day as they are “afraid of war”.
“People are in chaos, people are leaving” as they don’t know what is going to happen, he said.
Adama Barrow won the election and an inauguration is planned for Thursday.
But President Yahya Jammeh rejected the result and is refusing to step down until the Supreme Court hears his challenge.
His legal team has asked for an injunction to block Mr Barrow’s inauguration – something a Gambian court is expected to rule on on Monday.
Parliament has also reconvened for an emergency session. While the reasons for the meeting are unclear, the BBC’s Umaru Fofana in Banjul says that parliament may be asked to extend Mr Jammeh’s term of office.
Mr Barrow will stay in Senegal until his inauguration, state media in Senegal say.
The Gambia, a tiny country of less than two million people, is surrounded on three sides by Senegal and regional bloc Ecowas has said it is considering military intervention to force Mr Jammeh to relinquish power.
One woman boarding the boat with her three children told our correspondent that the political uncertainty made it hard to ensure she could look after her children.
“We don’t know what to expect with electricity outage, water shortage, food supplies,” she said.
Another said she was going “because things are not safe”.
“I’m going because I am afraid. I’m going until we have peace and then we can come back.”
The UN refugee agency said last week that several thousand people, mainly children, have crossed into Senegal from The Gambia since 3 January.
“UNHCR teams report seeing buses filled with children, accompanied by women, cross the border,” said Liz Ahua, the regional representative for UNHCR.
Mr Jammeh’s attempt to overturn the election result has been delayed because of a shortage of judges.
The African Union has said it will no longer recognise Mr Jammeh’s authority after his term ends.
The 51-year-old leader seized power in the country in 1994 and has been accused of human rights abuses, although he has held regular elections.