Georgian security forces are attempting to capture the Monadire (Hunter) militia of Emzar Kvitsiani, a former regional governor. The operation had been briefly halted because of darkness and fog.
The central authorities charged him with treason on July 23 after he refused an order to disarm.
Giorgi Arveladze, the head of the administration of President Mikheil Saakashvili, said that all villages in the gorge are under the control of security forces but that the hunt for Kvitsiani was continuing. He said some members of Kvitsiani's militia were detained by security forces, while others gave themselves up.
Reports from the region previously said government troops had surrounded Kvitsiani and about 60 of his men in mountains near the village of Azhara.
An eyewitness from the gorge told the Imedi television channel that government forces had bombed the village of Chkhalta, where some of Kvitsiani's men are thought to be hiding, where the woman was killed.
Kvitsiani says he is defending the rights of the local Svan people.
Up to 1,000 Georgian troops have been reported deployed in the area.
(AFP, Rustavi 2, Imedi TV, Caucasus Press, Novosti-Gruziya)