Reuters could not verify either side's assertions.
Vladimir Saldo, the Kherson region's Russian-installed governor, said the drones had struck a market in the town of Oleshky about 9.30am on what was a public holiday.
He and two social media users released videos purporting to show some of the incident.
Reuters confirmed the location as Oleshky but could not independently verify when they were filmed.
One of the videos showed an explosion going off beside one in a group of low-slung buildings.
Another video shows at least two bodies lying on the ground beside the structure; it was not possible to tell who they were.
Vladyslav Voloshyn, a spokesman for the Ukrainian military, said Ukraine had targeted Russian troops in part of the Kherson region and that only military personnel, and not civilians, had been killed.
Russia's foreign ministry called on the international community to condemn what it said was an act of terrorism.
"After the first strikes, which had already resulted in casualties, the enemy launched a second drone raid, during which they mercilessly killed the survivors and attacked the rescuers who had arrived," the ministry said in a statement.
"This is yet another bloody war crime."
Local Russian-installed official Sergei Cherevko said about 30 Ukrainian drones had struck the market in four different attack waves, according to state media.
Cherevko alleged Ukrainian forces had used cluster munitions against ambulances gathering at the scene and said two people had been killed - contradicting the death toll of seven given by Saldo and the Russian Foreign Ministry.
Reuters has requested clarification from Saldo.
Russia and Ukraine have accused each other of deliberately targeting civilians in the course of the war, now well into its fourth year, something both sides deny.
Civilian casualties have been far higher on the Ukrainian side.
Russian strikes last month which the Kremlin said had been aimed at military targets killed 19 people in the city of Kryvyi Rih, 35 in Sumy and 12 in the capital Kyiv, prompting US President Donald Trump to urge Russian President Vladimir Putin to "STOP!".
Rodion Miroshnik, a special ambassador for Russia's foreign ministry, called the strike on Oleshky a "monstrous atrocity" that made a mockery of Ukrainian President Volodymyr Zelenskiy's talk about peace.
Ukraine says it wants a ceasefire of at least 30 days to open the way for talks on securing a just peace.
Russia has not agreed to such a truce but Putin has unilaterally announced a three-day ceasefire next week and the Kremlin says he is open to direct talks with Ukraine.
