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.
Latest Stories
-
More money, less taxes, and smarter spending: Finance Minister breaks down new economic reforms
31 seconds -
Black Sherif unleashes highly anticipated sophomore album ‘Iron Boy’
7 minutes -
Kwame Agbodza criticises late award of 80km Sampa-Jinijini road contract
9 minutes -
Asiedu Nketiah struggles to gain ground in NDC’s 2028 presidential race – Global InfoAnalytics
11 minutes -
I am excited about Pencils of Promise’s intervention projects – Ja Rule
28 minutes -
Mahama’s job approval hits 66%; signalling strong confidence in him – Global InfoAnalytics survey
33 minutes -
Presidency and Military ranked least perceived corrupt institutions in Ghana now – Global InfoAnalytics
39 minutes -
Police, Immigration, Judiciary ranked top 3 most corrupt institutions in Ghana – Global InfoAnalytics survey
46 minutes -
Bawumia leads NPP race with 48% of Ghanaians approval rate – Global InfoAnalytics
52 minutes -
68% of voters say Akufo-Addo caused NPP defeat – Global Info Analytics
1 hour -
57% of NPP supporters prefer Bawumia to run for 2028 – Global InfoAnalytics
1 hour -
Nigerian pastor acquitted of rape after eight years in South African jail
2 hours -
Alcohol makes male fruit flies more attractive
2 hours -
UK to keep pushing for deal after Trump imposes 10% tariff
2 hours -
World leaders call Trump tariffs ‘wrong’ and ‘unjustified’
3 hours