The official said the bombing targeted a base belonging to a local militia group called February 17 and the area of Sidi Faraj, where members of the extremist organization Ansar al-Shariah are located.
An official from Ansar al-Shariah said no one was hurt in the bombing and claimed that one of two rockets fired by the aircraft hit a car dealership.
Both officials spoke on condition of anonymity because they were not authorized to speak to journalists.
General Khalifa Hifter has vowed to crush the Islamists and many military units back him.
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Hifter claims to have more than 75 per cent of Benghazi, Libya's second-largest city, under his control. He also says he is getting help from moderate Islamists who were breaking away from their militias and joining his forces.
The eastern city was the birthplace of the 2011 uprising that toppled dictator Moammar Gadhafi.
The 70-year-old Hifter is based in Benghazi, where he has been leading an armed revolt since late last month in what is perhaps the biggest challenge yet to Libya's weak central government and fledgling security forces since Gadhafi's ouster.