Ukrainian President Volodymyr Zelenskyy said the destruction of the Nova Kakhovka dam had restricted "hundreds of thousands of people" access to clean drinking water.
In a post on Twitter, he wrote that the evacuation was ongoing.
"We are working at all levels of state and local authorities to rescue as many people as possible from the flooded areas," Zelenskky wrote and again blamed Russia for the disaster.
"Russia must be held accountable for this deliberate crime against people, nature, and life itself," he wrote.
Earlier, Ukraine's domestic security service said it had intercepted a telephone call which it said proved a Russian "sabotage group" blew up the dam.
It posted a one-and-a-half-minute audio clip of the alleged conversation on its Telegram channel. DW could not independently verify the recording, and Moscow did not immediately comment on its content.
Russia previously claimed that the dam was destroyed by Ukrainian shelling.
The Nova Kakhovka dam was breached on Tuesday, with both Moscow and Kyiv trading accusations of responsibility for its destruction. Some experts said it might have been due to wartime damage and neglect.
Flooding from the breach extends over 600 square kilometers on the Ukrainian-held side of the Dnieper River as well as the Russian-held side of the river, the governor of the Kherson region earlier said.
Ukrainian Interior Minister Ihor Klymenko said on the Telegram messaging app that four people had died and 13 people were missing in the Kherson region, and that one person had died in the Mykolayiv region.
A Russian-appointed official said eight people had died in Russian-held territory and more than 5,800 had been evacuated from their homes.
The German Red Cross (DRK) meanwhile has sent 13.3 tons of disaster aid to assist flood victims. Supplies include water and hygiene kits to address the increased risk of illness due to contaminated water.
"During floods, there is an increased risk of falling ill from contaminated drinking water and a lack of sewage treatment," DRK spokeswoman Rebecca Winkels told the German news agency dpa.
The supplies will be handed over to the Ukrainian Red Cross for distribution.
Here are some of the other developments concerning Russia's war in Ukraine on Friday, June 9:
Nuclear plant still getting water — IAEA
The breached Kakhovka dam is still supplying Europe's largest nuclear power plant with cooling water, the UN nuclear agency said on Thursday.
"Ukraine's Zaporizhzhia Nuclear Power Plant is continuing to pump cooling water from the Kakhovka reservoir," the International Atomic Energy Agency (IAEA) said in a statement.
It followed claims by the dam's operator to Ukrainian TV, saying the reservoir had gone "below the critical point of 12.7 meters (42 feet)."
If the claims were true, the reservoir would not have been able to supply Zaporizhzhia's ponds, which are needed to cool the plant.
The IAEA said an assessment by its experts indicated that the plant's pumps could likely continue to be operated even at the level of 11 meters or lower.
"In these difficult and challenging circumstances, this is providing some more time before possibly switching to alternative water supplies," IAEA head Rafael Grossi said.
However, Grossi, who is due to visit the plant next week on a third visit since the war started, warned of the "very precarious and potentially dangerous" safety and security situation around the plant, amid the continuing fighting.
Russia says no grounds to extend grain deal — report
Russia's ambassador to Turkey said that while Moscow is continuing consultations with the UN regarding the Black Sea grain deal, there aren't any grounds to extend it, according to a report by Russian news agency RIA.
On Wednesday Russia accused Ukraine of an attack on an ammonia pipeline and threatened to end the agreement on shipping grain from Ukrainian ports because of it.
The grain agreement concluded last summer brought an end to a months-long Russian naval blockade of Ukrainian Black Sea ports.
Talks on the agreement are taking place in Geneva.
Russia reports heavy fighting in southern Ukraine
Russia's military leadership has reported heavy fighting in the Zaporizhzhia region in southern Ukraine, and claimed to have fought back repeated Ukrainian attempts to break through the front line.
President Vladimir Putin was briefed on the military's claims that a Ukrainian counter-offensive had been repelled.
Kyiv has remained tight-lipped on any mention of an expected counter-offensive and has accused Moscow of spreading lies.
Neither sides battlefield accounts can be independently verified.
Ukraine says one killed in Russia air strike
Ukraine's interior ministry said one person had been killed, and three were wounded in a Russian air strike overnight.
Ukraine's military said it shot down four cruise missiles and ten attack drones during the attack.
During an earlier attack, the air force also said two cruise missiles had struck a civilian object in the central Ukrainian region of Cherkasy.
Regional governor, Ihor Taburets, said at least eight people had been wounded in that strike.
Zelenskyy hails Donetsk fighting 'results'
Ukrainian President Volodymyr Zelenskyy has applauded the "results" of heavy fighting in eastern Ukraine's Donetsk region.
Zelenskyy spoke during his daily video message, which he delivered late on Thursday aboard a train after visiting areas affected by the Kakhovka dam breach.
The president acknowledged the heavy fighting in the Donetsk region.
"But there are results, and I am grateful to those who achieved these results. Well done in Bakhmut. Step by step," he said.
US to announce fresh $2 billion arms package — reports
The United States is preparing to announce a fresh package of arms support to Kyiv worth $2 billion (approximately €1.86 billion), Bloomberg News reported late on Thursday, citing administrative officials.
The funds will be presented under the Ukraine Security Assistance Initiative, Bloomberg said.
The package will include a generous amount of air defense munitions. It's also meant to help Ukraine purchase Hawk missile launchers, alongside two types of advanced Patriot air defense missiles.
Biden, Sunak stress 'unwavering support' for Ukraine
US President Joe Biden and British Prime Minister Rishi Sunak have stressed their "unwavering support" for the people of Ukraine in their fight against the Russian invasion.
The two leaders spoke at the White House on Thursday, during Sunak's visit to Biden.
"The UK and the US together with more than 50 partners have committed historic levels of security assistance to Ukraine," Biden said after the talks.
London and Washington are two of the biggest donors to Ukraine, and they play a central role in a long-term, recently announced effort to train Ukrainian pilots on F-16 fighter jets.
