Ukrainian cities have also been struck in recent days by drones and missiles.
India on Wednesday issued an advisory asking all its citizens to avoid travelling to Ukraine, citing the "deteriorating security situation" in the country.
"In view of the deteriorating security situation and recent escalation of hostilities across Ukraine, Indian nationals are advised against travelling to Ukraine," the Indian embassy in Ukraine said on its Twitter handle.
"The Indian citizens, including students, currently in Ukraine are advised to leave Ukraine at the earliest by available means," it added.
The advisory came as Russian President Vladimir Putin introduced martial law on Wednesday in four Ukrainian regions he says Russia has annexed, while some residents of the occupied city of Kherson left by boat following warnings of a looming assault.
The images of people fleeing Kherson were broadcast by Russian state TV which portrayed the exodus - from the right to left bank of the River Dnipro - as an attempt to clear the city of civilians before it became a combat zone.
Kirill Stremousov, deputy head of the local Russia-backed administration, made a video appeal after Russian forces in the area were driven back by 20-30 km in the last few weeks. They risk being pinned against the western bank of the 2,200-km-long Dnipro river that bisects Ukraine.
In a move which looked designed to help Russia firm its grip on the Ukrainian regions it partly occupies - including Kherson - Putin told his Security Council he was introducing martial law in them.
Beyond much tighter security measures on the ground, it was unclear what the immediate impact of that would be.
Kyiv, which does not recognise Moscow's self-styled annexations of the four regions, derided the move.
"'Martial law' implementation on the occupied territories by Russia should be considered only as a pseudo-legalisation of (the) looting of Ukrainians' property," tweeted Mykhailo Podolyak, a Ukrainian presidential adviser.
"This does not change anything for Ukraine: we continue the liberation and deoccupation of our territories."
Eight months after being invaded, Ukraine is prosecuting major counter-offensives in the east and south to try to take as much territory as it can before winter after routing Russian forces in some areas.
The conflict has killed thousands, displaced millions, pulverised Ukrainian cities, shaken the global economy and revived Cold War-era geopolitical fissures.
Putin also issued a decree restricting movement in and out of eight regions adjoining Ukraine and ordered the creation of a special coordinating council under Prime Minister Mikhail Mishustin to step up the faltering war effort.
Kherson is the biggest population centre Moscow has seized and held since it began its "special military operation" in Ukraine on Feb. 24.
Andriy Yermak, head of the Ukrainian president's office, accused Russia of laying on a propaganda show there.
"The Russians are trying to scare the people of Kherson with fake newsletters about the shelling of the city by our army, and also arrange a propaganda show with evacuation," Yermak wrote on the Telegram messaging app.
'LOOMING OFFENSIVE'
Ukrainian cities have also been struck in recent days by drones and missiles, and Vitali Klitschko, Kyiv's mayor, said the capital's air defences were in action once again on Wednesday.
Ukrainian President Volodymry Zelenskiy, who has said a third of his country's power stations have been hit by Russian strikes, on Wednesday discussed security at power supply facilities with senior officials.
"We are working to create mobile power points for the critical infrastructure of cities, towns and villages," Zelenskiy wrote on the Telegram messaging app.
"We are preparing for various scenarios of possible consequences. Ukraine will defend itself. No matter what the enemy plans and does."
In Kherson, Stremousov said the city and especially its right bank could be shelled by Ukrainian forces, adding that residents who left would be given accommodation inside Russia.
"I ask you to take my words seriously and to interpret them as a call to evacuate as fast as you possibly can," he said.
"We do not plan to surrender the city, we will stand until the last moment."
The Russian-installed chief of Kherson region, Stremousov's boss, said about 50,000-60,000 people would be evacuated in the next six days. The city of Kherson had a pre-war population of around 280,000 people but many of them have since fled.
"The Ukrainian side is building up forces for a large-scale offensive," Vladimir Saldo, the official, told state TV. "Where the military operates, there is no place for civilians."
Saldo, who said Russia had the resources to hold Kherson and even counter attack if necessary, also said he was banning civilians from entering the region for seven days.
Staff at Kherson's Russian-backed administration were also being relocated to the left bank of the Dnipro, he said.
The evacuation calls followed a gloomy assessment of Russia's prospects in the area from General Sergei Surovikin, the new commander of Russian forces in Ukraine.
"The situation in the area of the 'Special Military Operation' can be described as tense," Surovikin told state-owned Rossiya 24 news channel. "The situation in this area (Kherson) is difficult. The enemy is deliberately striking infrastructure and residential buildings."