Ukraine has been trying to rescue people trapped on the Russia-held eastern bank of the Dnipro River since the Nova Kakhovka dam was destroyed.
Oleksandr Prokudin said Russian troops shelled the evacuation boat and shot a 74-year-old man dead.
The man was trying to rescue a woman from gunfire, Mr Produkin said.
Two police officers were also injured.
The Nova Kakhovka dam burst on Tuesday, releasing a huge torrent of water which quickly flooded vast areas of land on both sides of the Dnipro river.
Ukraine has blamed Russia for "blowing up" the dam, located in a Russian-controlled area of Ukraine. Russia has denied this and has accused Ukraine of being responsible for its destruction.
The BBC has not verified either claim, although it appears likely that Russian forces, who controlled the dam, decided to blow it up in order to complicate Ukraine's ongoing counteroffensive.
The eastern bank of the Dnipro River has been one of the areas worst hit by the flooding, with hundreds of people there posting on the Telegram app asking to be rescued.
Ukraine's military says it has been co-ordinating rescues from the eastern bank, but claimed "fearless volunteers" were carrying out some of the evacuations.
One of those involved in the rescue effort, Viktor, told the BBC he came under Russian fire while attempting a trip, saying Russian soldiers were "waiting for volunteers or soldiers to arrive so they can shoot them".
The BBC has not been able to independently verify these claims.