It was the sixth drone attack on the capital this month. At least two people were injured.
Klitschko said a drone had hit a block of flats in the Solomyanskyi district, south of the city centre, early on Friday triggering a fire.
Two people were injured, including one being treated in hospital.
"The upper floors are in flames. Emergency crews are on site," Mayor Klitschko said.
A video posted on social media showed a giant orange flame going skyward in the night.
Klitschko also said drone fragments had set fire to a house under construction in Darnytskyi district on the eastern bank of the Dnipro River that runs through the city.
He said there were no injuries. Pictures posted online showed construction materials strewn about the site.
Serhiy Popko, head of Kyiv's military administration, reported fragments from a downed drone had struck an apartment building in a third area - Holosiivskyi district - also south of the city centre.
Popko posted pictures showing smashed windows and heavy damage to apartments.Â
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