The city's military administration said seven sites had come under attack in the assault that began after midnight.
Air raid alerts for Kyiv remained in place for more than an hour.
Mayor Vitali Klitschko, writing on Telegram, said one warehouse building had been destroyed near the city centre and rescue teams had pulled two people from the ruins.
"There may still be people under the rubble," he wrote. "Search and rescue operations are proceeding."
He said an ambulance driver was among the injured.
Klitschko said the attack triggered fires in warehouses and storage areas, but initial reports that a 20-storey apartment building was ablaze proved to be inaccurate.
A large fire broke out on the city's outskirts, he said, and debris from a fallen missile had landed next to a residential building.
The city's military administration also said the attack triggered an ammonia leak, which was being tackled by emergency crews.
Reuters witnesses said explosions had been heard throughout the city. Klitschko said air defence units had gone into action to repel the assault. In the surrounding Kyiv region, another person was killed and 21 other people injured, local authorities said on Telegram.
Russia has intensified its attacks on Kyiv in recent weeks, with several assaults on the capital in the past month. Reuters could not independently verify all the reports.
Earlier, Ukraine accused Russia of committing a war crime after a video on social media showed a drone chasing down and wounding a vegetable seller in the southern Ukrainian city of Kherson.
Footage shared by Ukrainian police shows the drone pursuing a man around a van being used as a street stand before exploding as he dives for cover behind the vehicle.
Police said the man in his 50s survived but suffered shock and shrapnel injuries.
The video appeared on a Telegram channel in Kherson and was later shared by the local police, although its origin remains unclear.
Ukrainian officials later identified him only as a 52-year-old called Yurii and released a video interview with him from a hospital ward.
"Today, many were shocked by yet another video of the drone 'safari' Russians are carrying out against civilians in Kherson," President Volodymyr Zelenskiy wrote on X, saying that "Russia has gone mad" and calling for strong security guarantees for Ukraine.
"We must act and support life in ways that force Russia to think seriously about peace rather than expanding the range of its drones."
Reuters verified the location using buildings, the road layout and vegetation that matched satellite and archive imagery.
The date could not be independently confirmed but Ukraine's police and Foreign Minister Andrii Sybiha said the incident happened on Tuesday.
The vegetable seller went to Kherson in the morning to sell produce, said Oleksandr Prokudin, head of the Kherson regional military administration.
Prokudin relesed a video interview with the man on Telegram.
"I only just put up the umbrella and started unloading the boxes. My wife was there, too. Then we heard it buzzing but could not even see it. It came from somewhere behind the market, from those bushes over there, and flew in very low," said Yurii, sitting in a hospital ward.
"It is open steppe. There is nowhere to go. It is terrifying, of course. Really terrifying."
Speaking from a hospital bed, the man said he tried to signal to the drone operator that he was only selling tomatoes and aubergines.
Sybiha said the attack formed part of a deliberate Russian campaign to intimidate civilians.
with DPA
