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 the government's headquarters.
They also clashed with security forces in Gen Khalifa Haftar's stronghold of al-Marj for the first time.
The UN's Libya mission "expresse[d] grave concern" over reports that a civilian was killed.
AFP also reports that officers opened fire on protesters who forced their way into a police station, injuring five people.
A spokesman for Gen Haftar said the administration backed peaceful protests but would not allow "terrorists and the Muslim Brotherhood" to hijack them.
The parliament in Tobruk needs to approve the resignation of the government.
Libya has been torn by violence since long-time ruler Col Muammar Gaddafi was deposed in 2011 by Nato-backed forces.
Until now, protests against the situation in Libya have focused largely on the capital, Tripoli, home to the UN-recognised Government of National Accord (GNA).
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.
In early May, a leaked UN report said that hundreds of mercenaries from the Wagner Group - run by Yevgeny Prigozhin, a close associate of Russian President Vladimir Putin - were operating in Libya.
Gen Haftar is also backed by the United Arab Emirates and Egypt, while the GNA enjoys the support of Turkey, Qatar and Italy.
Libya has the biggest reserves of oil and gas in Africa.
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.
Libyans losing patience with their leaders
By BBC North Africa correspondent Rana Jawad
Libya's parallel government in the east of the country is not recognised by the international community.
It has broadly existed for years as the political arm - and some would argue cover - for Gen Haftar's self-proclaimed army which controls that region.
Its spokesman has said they would protect demonstrators, and the quick reaction could be seen as a sign of nervousness in their camp.
It is the worn-out patience of civilians amid Libya's continued instability that poses the most serious challenge to all armed and political bodies, rather than the resignation of any one bloc.
Protests are being triggered by worsening living conditions including severe power cuts, an ongoing banking crisis, and what civilians see as widespread corruption.