* Fifth day of fighting between rival militias
* Air controllers halt work, deepening Libya's isolation
* Eastern politician is second woman to be assassinated (Adds details, background)
By Hani Amara and Ahmed Elumami
TRIPOLI/BENGHAZI, Libya, July 17 (Reuters) - Several shells hit the terminal of Libya's main airport on Thursday as rival militias fought in Tripoli for a fifth day, and gunmen assassinated a female politician in the east.
In another sign of growing turmoil, air controllers halted work in Tripoli, shutting off much of the oil-producing country from international traffic.
Tripoli International Airport has been a battlefield since fighters attacked it with heavy guns on Sunday to wrest control from a rival militia which has been based there since the fall of Libya's late ruler Muammar Gaddafi in 2011.
The conflict is fuelling worries that Libya is on the point of turning into a failed state where a weak central government is powerless to control the militias which helped oust Gaddafi.
The airport fighting pits brigades from Misrata, a western coastal town, against rival fighters from Zintan in the northwest. Their rivalry exemplifies the divisions between tribes and cities in a country where few efficient state institutions exist after over four decades of one-man rule.
On Thursday, several shells hit the airport terminal where the Zintanis are holding out, striking the main building for the first time, witnesses said.
A Reuters reporter at the airport saw holes in the roof and smashed windows at the terminal building and in airline offices, including one belonging to British Airways, with a shell lying on the floor.
Air controllers refused to go to work at the control tower in Tripoli, which regulates traffic for all of western Libya, a spokesman for the transport ministry said.
On Wednesday, Libya reopened the western Misrata airport, which had been closed with Tripoli after the weekend attack, but it will have to shut again because Tripoli air controllers are also responsible for Misrata.
Many Libyans who had been planning to come home for the Muslim fasting month of Ramadan have been trapped abroad. Expatriates trying to leave the country have been travelling by taxi to Tunisia, in scenes reminiscent of the 2011 NATO-backed uprising.
In the eastern Islamist hotspot of Derna, gunmen shot dead Fariha al-Barkawi, a former member of parliament, officials said. She is the second prominent woman to be assassinated, following the killing of Benghazi human rights activist Salwa Bugaighis last month.
Western powers fear chaos in Libya will allow arms and militants to flow across its borders. The south of the vast desert country has become a haven for Islamist militants kicked out of Mali by French forces earlier this year. (Reporting by Hani Amara, Ahmed Elumami and Ulf Laessing; Writing by Ulf Laessing; Editing by Mark Trevelyan)