Witnesses say unrest has spread to the capital Yamoussoukro, and the UN has warned that the country is at risk of relapsing into civil war.
Mr Gbagbo refused to stand down after losing an election last November.
The UN and foreign leaders have urged Mr Gbagbo to hand power to rival Alassane Ouattara, widely seen as the winner of the election.
The country has been divided along north-south lines since the end of civil war in 2003.
Much of the recent fighting has centred on the main city of Abidjan, which is in Gbagbo-controlled territory.
This week, there has been heavy fighting in the Abobo neighbourhood of Abidjan, forcing hundreds of residents to flee.
The BBC's West Africa correspondent Thomas Fessy says it is unknown how many people have been killed during the gun battles, but Abobo is now the scene of daily bloodshed.
Meanwhile, a resident of Yamoussoukro, north-west of Abidjan, told the BBC that shots had been fired in the streets of the capital overnight.
Unconfirmed reports said five people were killed after clashes between pro-Gbagbo soldiers and Ouattara supporters.
It comes a day after violence erupted in the west of the country.
After fighting overnight, the rebels took the small western town of Zouan-Hounien from a pro-Gbagbo militia, residents and a militia leader said.
"Zouan-Hounien has fallen to the rebellion but we're in the process of re-organising ourselves [with the Ivorian army] to retake the town," Yao Yao, head of operations of the Front for the Liberation of the Greater West militia, told Reuters news agency.
On Friday, UN secretary-general Ban Ki-moon expressed concern at the spreading violence.
"These developments mark a disturbing escalation which draws the country closer to the brink of reigniting civil war," Mr Ban said in a statement released by his office.
He called on pro-Gbagbo forces to stop intimidating UN peacekeepers in the country.
Separately, the UN's refugee agency says the number of civilians fleeing west to Liberia has surged.
"Until mid-week we were seeing around 100 people crossing the border daily. But over the past 24 hours alone, the numbers coming across have swollen to 5,000 people," the agency said in a statement.
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