Ali Ahneesh, head of the Red Crescent branch in Zawiya, told the Istanbul-based Libya Alahrar TV channel that 10 families had been evacuated, and called for “a ceasefire to evacuate families stuck in the areas where the clashes have taken place”.
Red Crescent volunteers had been receiving calls from families in the conflict area asking to be evacuated, he said.
There was no immediate indication of who had taken part in the violence or why they were fighting.
Imad Ammar, a member of Zawiya's elders and notables council, said the fighting appeared to involve individuals rather than armed groups.
Zawiya, 40 km (25 miles) west of the capital Tripoli, is home to Libya's biggest functioning refinery, with a capacity of 120,000 barrels per day.
"The clashes in the morning were fierce, and the casualties are one killed and six injured," Tripoli-based Ambulance and Emergency Services spokesperson Osama Ali told the TV channel.
Ali said rescue teams had been unable to reach the conflict zone, and it was not clear if the casualties were civilian or military.
Zawiya has witnessed repeated armed clashes that have at times forced the closure of the coastal road to the border with Tunisia.
SD/PR