Adama Barrow, who beat him in elections last month, is due to be inaugurated as the new president today, but Mr Jammeh has ignored the deadline.
West African military forces are ready to enforce a transfer of power., including Nigeria sending down soldiers.
Senegalese troops remain stationed at the Gambian border, despite the deadline for Mr Jammeh to stand down passing at midnight.
The threat of military action is supported by Nigeria and other states in the region.
However, Gambian army chief Ousman Badjie said his troops would not fight Senegalese forces should they enter into the country, AFP news agency was quoted to have said.
"We are not going to involve ourselves militarily, this is a political dispute," he said. "I am not going to involve my soldiers in a stupid fight. I love my men."