Senegalese troops have been seen moving towards the Gambian border in a show of force to pressure President Yahya Jammeh to stand down.
Senegal gave him a midnight GMT deadline to quit and Nigeria has sent an air force unit to Senegal in support of the possible intervention.
Wednesday was meant to be his last day in office but parliament has granted him three more months in the post.
It effectively stops successor Adama Barrow being sworn in on Thursday.
His shock victory in the 1 December election plunged The Gambia into crisis.
West African countries are seeking UN backing to intervene militarily to eject Mr Jammeh, who has ruled The Gambia since taking power in a bloodless coup in 1994.
Meanwhile, thousands of UK and Dutch tourists are being evacuated from the tiny West African state, which is popular with European holidaymakers because of its beaches.
Why is Senegal taking the lead?
Ecowas, the Economic Community of West African States, has mandated Senegal, which almost surrounds The Gambia, to spearhead military intervention, but only as a last resort and with the backing of the UN Security Council.
The Gambia’s entire armed forces is made up of only about 2,500 troops, making it difficult to see how they can defeat a regional force if it moves in, says BBC Africa Monitoring security correspondent Tomi Oladipo.
Mr Jammeh has declared a 90-day state of emergency, saying “any acts of disobedience to the laws of The Gambia, incitement of violence and acts intended to disturb public order and peace” were banned.
Security forces, he said, had been instructed to “maintain absolute peace, law and order”.
Could there be a refugee crisis?
At least 26,000 Gambians, mostly women and children, had crossed into Senegal by Monday evening amid fears that violence could erupt, the UN refugee agency said, citing Senegalese government figures.
“The flow has increased sharply since then,” regional spokeswoman Helene Caux was quoted by Reuters as saying.
Travel firm Thomas Cook said it would fly back to the UK, over the next 48 hours, 985 customers from package holidays after the Foreign Office advised “against all but essential travel to The Gambia due to ongoing political uncertainty and potential military intervention following the presidential elections”.
About 1,600 Dutch citizens are also being flown home after similar advice from their government.
Tourism has become the fastest-growing sector of The Gambia’s economy, and the country, which has a population of about two million, was marketed to holidaymakers as “the smiling coast of West Africa”.
But many of its citizens are poor and complain of political repression. Some, including the goalkeeper of the national women’s football team, have drowned in the Mediterranean trying to reach Europe in the hope of a better life.
Why is Mr Jammeh refusing to leave office?
The Gambia regularly held elections, which he won until last year.
Mr Jammeh has said there were irregularities in the election process, including the turning away of some of his supporters from polling stations, and errors made by the electoral commission.
The commission accepted that some of the results it initially published contained errors, but said Mr Barrow had still won.
Mr Jammeh has said he will stay in office until new elections are held.
Retaining power would also ensure he was not prosecuted in The Gambia for alleged abuses committed during his rule.
The US state department urged Mr Jammeh to peacefully transfer power to Mr Barrow on Thursday.
“Doing so would allow him to leave office with his head held high and to protect The Gambian people from potential chaos,” spokesman John Kirby said.