More than 45,000 have fled country this month - UN
Jammeh, who has been in power more than 20 years, lost elections last month to Adama Barrow but has refused to stand down.
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"Around 45,000 people are now reported to have arrived in from The Gambia... amid the ongoing political uncertainty as Senegalese and West African troops entered the country on Thursday," the UN refugee agency said in Geneva.
"It is feared that more people may continue to flee as the situation remains tense," UNHCR said, adding that another "800 people have crossed into Guinea-Bissau."
The UN agency noted that arrivals in Senegal have included "Gambians, Senegalese, bi-nationals, as well as Ghanaians, Liberians, Lebanese, Guineans, and Mauritanians among other foreigners."
Soldiers from Senegal and four other west African countries have crossed into The Gambia to bolster Barrow, who has been sworn in at the country's embassy in Dakar.
West African leaders have said regional forces would intervene with force if Jammeh refused to stand down.
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