The hit on the regional capital, some 90 kilometres (55 miles) from the Ukrainian border, comes two days after a Russian ballistic missile killed at least 35 people in the Ukrainian city of Sumy.
Aerial attacks have escalated despite calls by US President Donald Trump for a ceasefire to halt three years of fighting.
Kursk Governor Alexander Khinsthein called the strike a "crime of unprecedented cruelty," in a post on Telegram, and said an 85-year-old woman had died.
Six of the nine injured were hospitalised for shrapnel wounds, burns, and head injuries, the Kursk authorities said.
Russia's defence ministry said Ukraine fired 115 drones at its territory overnight, with 109 shot down over the Kursk region.
Strikes on villages either side of the Russia-Ukraine border have become a daily occurrence in the conflict, though deaths as a result of Ukrainian hits on Russian cities and areas further from the front lines are less frequent.
Russian investigators said the drones had been packed with metal shrapnel "to cause maximum damage to civilians and infrastructure".
The regional government posted photos on social media showing damage to an apartment block and debris strewn on the ground.
A Kyiv official alluded to the attack on social media but claimed a military target had been hit.
"Unknown drones hit a Russian ammunition depot in Kursk," the head of disinformation countering centre Andriy Kovalenko said on Telegram.
Russia has pushed Ukraine out of all but 50 square kilometres of the Kursk region, according to an AFP analysis of Institute for the Study of War data.
And Moscow's army has stepped up pressure on the regions of Sumy and Kharkiv this year, claiming several border villages amid warnings from Ukraine's army of a fresh Russian offensive there.
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