The latest protests erupted on Saturday evening
The rival government in eastern Libya has submitted its resignation after a number of protests over deteriorating living conditions and corruption.
At the weekend demonstrators in the city of Benghazi set fire to strongman Gen Khalifa Haftar's headquarters.
They also clashed in his stronghold of Al-Maj for the first time.
Libya has been torn by violence since long-time ruler Col Muammar Gaddafi was deposed and killed in 2011 by Nato-backed forces.
A spokesman for Gen Haftar said the administration backed peaceful protests but would not allow "terrorists and the Muslim Brotherhood" to hijack them.
Until now, protests against the situation in Libya have focused largely on the capital, Tripoli, home to the UN-recognised Government of National Accord.
Gen Haftar's forces launched an offensive to seize Tripoli in April 2019 but military support from Turkey helped government forces drive Gen Haftar's Russian-backed troops back from the frontlines earlier this year.
Since January an armed group loyal to Gen Haftar has blocked key oil fields, causing power cuts and costing Libya billions of dollars in lost exports.
It was partially lifted last month to allow the sale of oil already stored in terminals, but not the production of fresh supplies.
