President Jammeh has blamed dissidents based in the US, Germany and the UK for the attack.
Heavy gunfire was heard near the presidential palace in Banjul on Tuesday but details are sketchy.
Mr Jammeh seized power in the tiny West African nation in 1994, and is accused of not tolerating any opposition.
The president denied that the attack was mounted by sections of the Gambian military.
"The Gambian armed forces are very loyal as far as we are concerned - there isn't any single participation of the armed forces except nullifying the attack," he said in a television interview on Wednesday night.
"So this cannot be called a military coup - this was an attack by a terrorist group backed by some powers that I would not name."
The US government has denied that it had any role in the alleged coup attempt.
Mr Jammeh also warned that he would make "an example" of anyone who had been part of the attack.
It is not clear how many people have been arrested but sources have told the BBC that several relatives of alleged coup leader Lt Col Lamin Sanneh have been detained.
Lt Col Sanneh is among five people said to have been killed during the attack but this has not been confirmed.
He was a former head of the presidential guards who fled to the US after falling out with the president.
Gambian state TV showed a haul of weapons said to have been seized during Tuesday's fighting.
UN chief Ban Ki-moon has called for an investigation into the attack and also urged the government and security forces in Banjul to "act in full respect of human rights".
Mr Jammeh was out of the country at the time of the attack but has since returned to the capital Banjul.
He has won several elections since seizing power but critics say they were neither free nor fair.
Numerous officials and opposition leaders have fled the country, saying they fear for their lives.
Apart from a short Atlantic coast, The Gambia is entirely surrounded by Senegal.
The main foreign-currency earners for the country of two million people are tourism and peanut exports.
Soldiers loyal to Gambian President Yahya Jammeh are rounding up opponents after a reported coup attempt, sources say.