Ukraine’s army chief Valeriy Zaluzhnyi said he could not yet confirm whether Russian units were fully retreating across the Dnipro River to its east bank.
But he said that Ukrainian troops had advanced seven km (4.3 miles) in the past 24 hours and recaptured 12 settlements.
“We continue to conduct the offensive operation in line with our plan,” he wrote in a post on Telegram.
A small group of Ukrainian soldiers was shown on Ukraine’s state TV in the centre of the village of Snihurivka roughly 5km (34 miles) north of Kherson city.
They were greeted by dozens of residents in a square, with a Ukrainian flag fluttering from a pole behind them.
Moscow ordered its troops on Wednesday to withdraw from the entire Russian-held pocket on the west bank of the Dnipro River, including Kherson city, the only regional capital Russia had captured in nine months of war.
Ukrainian officials have so far mostly been wary in public, warning that Russians may still be planning to sow destruction on their way out or be laying a trap.
Mykhail Podolyak, an adviser to President Volodymyr Zelensky, said on Thursday Russia wanted to turn Kherson into a “city of death”, mining everything from apartments to sewers and planning to shell the city from the other side of the river.
Britain’s Defence Secretary Ben Wallace said exiting the southern city would be a “significant psychological blow” for Mr Putin’s troops, but warned “we will believe it when we see it”.
In a speech in Edinburgh, at a meeting of allies in the Joint Expeditionary Force grouping, he questioned what the “tens of thousands of deaths” were for following the failure in Russia’s objectives.
“It must be quite a significant psychological blow that the one objective they did manage to capture, they have announced their intention to leave,” he said.
“Of course this is Russia, so we haven’t yet seen them leave en masse.
“We will believe it when we see it and I think we should all be cautious, as President Zelensky was, that there is still Russian tricks and all sorts of things.”
Rishi Sunak said Russia’s withdrawal from Kherson would further demonstrate the weakness of Mr Putin’s military offensive, after he was forced to abandon his lightning invasion plan to seize Kyiv within days.
In a call on Thursday morning with Mr Zelensky, the Prime Minister expressed careful optimism over Moscow’s troops being forced to flee Kherson.
A Downing Street spokesman said: “The leaders agreed that any Russian withdrawal from the occupied city of Kherson would demonstrate strong progress for the Ukrainian forces and reinforce the weakness of Russia’s military offensive, but it was right to continue to exercise caution until the Ukrainian flag was raised over the city.
“The Prime Minister praised the bravery of the Ukrainian armed forces and reiterated the UK’s unwavering military, economic and political support.”
Russia denies it targets civilians despite bombarding residential areas throughout the conflict. It has been evacuating thousands of civilians from the Kherson area in recent weeks in what Ukraine calls a forced deportation.
At the front north of Kherson there was less artillery audible than usual. A heavy fog had settled in overnight, light snow fell and the ground was coated with frost.
Ukraine wants to inflict as much damage as possible on the Russian troops as they try to escape across the river, and Kyiv’s public caution may in part reflect its policy of maintaining silence about its own operations.
Russia’s retreat order, just over a month after Mr Putin proclaimed the annexation of an area he said would be part of Russia forever, is one of the most humiliating defeats Moscow has suffered so far.
Defence Minister Sergei Shoigu was shown on TV on Wednesday giving the retreat order in response to advice from his top commander who said it was necessary to save the lives of troops who would be better able to defend the Dnipro’s opposite bank.
General Mark Milley, chairman of the US joint chiefs of staff, said initial indicators suggested Russia was following through with its withdrawal, though it could take time to complete.
President Joe Biden said the withdrawal showed there were “some real problems with the Russian military.”
Following Russian defeats in northern and eastern Ukraine, the retreat leaves Moscow with only limited gains to show for a “special military operation” that made it a pariah in the West and killed or wounded up to 100,000 Russian troops, according to US estimates, with a similar number of Ukrainian soldiers believed to have died, as well as around 40,000 civilians.
Russian forces are still holding on to other gains in the south, including a vital land route connecting Russia to the Crimea peninsula it seized in 2014, and cities in the east that they mostly obliterated while capturing them.
For Ukraine, which has endured nine months of bombardment and occupation, victory in Kherson would strengthen the case that it can defeat Russia on the battlefield, and may quieten some Western voices calling for it to negotiate a deal that would cede territory.
“This is a validation of Ukraine’s military strategy and the approach taken by its senior leadership. They are succeeding and the Russians know it,” tweeted Mick Ryan, a retired Australian general.
“Now is NOT the time to force Ukraine into negotiations. The Russians might be weakened but they are not giving up on their territorial aspirations. They will have to be beaten on the battlefield and pushed out of Ukraine.”