The Kherson prosecutor's office said two others were also injured, while the interior ministry said eight more were hurt by shelling in Korabelna Square.
The attacks came as President Volodymyr Zelensky visited the city, where he met with locals affected by the flooding.
2,000 people have been evacuated from the area, Kherson's governor says.
Speaking in a video statement posted to Telegram, Oleksandr Prokudin said the "evacuation from zones of flooding is continuing" despite the "immense danger and constant Russian shelling".
But he said that 68% of the flooded territory in the Kherson region was on Russian-held territory on the east bank of the Dnipro River.
The river has slowly swelled since the Kakhovka dam collapsed on Tuesday, causing thousands to flee their homes. The World Food Programme told the BBC on Thursday that the situation was a "public health crisis in the making" due to pollution including sewage, heavy oil and pesticides mixed into floodwater.
Ukraine says the flooding has affected an area of around 600 km sq (230 sq miles), and hundreds of thousands of people have been left without drinking water. The Ukrainian army has used drones to drop water bottles and food to some residents.
While the water level seems to have stabilised in Kherson itself, it still flows through streets at a daunting height, and flies now fill the air along with a pungent smell.
Rescue teams and volunteers are continuing to head out on boats to salvage anyone, or anything they can. Their efforts are punctuated by outgoing artillery fire.
Both Kyiv and Moscow have accused each other of targeting evacuation points in the Kherson region. Mykhailo Podolyak, a senior adviser to President Zelensky, accused Russia of bombarding the city and of "preventing rescuers from evacuating the population".
The Kremlin-installed head of the Kherson region, Vladimir Saldo, said on Telegram that two people died after Ukraine shelled a civilian evacuation point which was flooded after the dam breach.
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