In a statement posted on the Telegram messaging app, Kursk region governor Roman Starovoit said a woman had been killed by shelling in the village of Gornal, near the border with Ukraine.
At least five drones were shot down over the Kursk region on Tuesday, Starovoit and Russia's defence ministry said.
The governor of Oryol region, Andrei Klychkov, said two drones had hit a fuel facility, injuring three and causing a fire that was later extinguished. He said three drones had been shot shown.
In a statement posted on Telegram, Klychkov called the events "an enemy attack".
Later the governor of the nearby Bryansk region, Alexander Bogomaz, and the defence ministry reported the destruction of another drone, without providing any details.
Russian border regions have repeatedly come under fire from Ukraine in recent months. On Dec. 30, at least 20 people were killed in a missile strike on the city of Belgorod, 40 km (25 miles) from Ukraine, Russian media said.
Russia has also recently fired hundreds of missiles and drones at Ukrainian cities, including some far behind the frontline, Ukrainian officials have said.
In the latest strike this week, Russia fired 51 missiles of various types, killing at least four people and hitting civilian infrastructure, they said.
(Writing by Felix Light and Maxim RodionovEditing by Gareth Jones)
