The Ukrainian army entered Russia's Kursk region on August 6, capturing dozens of settlements in the biggest offensive by a foreign army on Russian soil since World War II.
Ukraine separately targeted four Russian airfields overnight with drones in the "largest attack" of its kind since Moscow invaded in 2022, a source in Kyiv's security services told AFP.
"In the Kursk region, we are moving further. From one to two kilometres (0.6-1.2 miles) in different areas since the beginning of the day," Zelensky said on social media.
He also said Ukraine had captured "more than 100 Russian servicemen" over the same period and that this would "speed up the return home of our boys and girls".
The neighbouring Russian region of Belgorod meanwhile declared a state of emergency, as the governor warned the situation there was "extremely difficult" due to Ukrainian shelling and drone attacks.
An AFP analysis of data provided by the Institute for the Study of War (ISW) indicated Ukrainian troops had advanced over an area of at least 800 square kilometres of Russia as of Monday.
Russia said it had repelled attempts by Ukrainian forces to push deeper into Kursk region in five areas.
"The attempts by enemy mobile units using armoured equipment to break through deeper into Russian territory have been repelled," its defence ministry said.
Ukraine said Tuesday it would not hold on to Russian land it captured and offered to stop raids if Moscow agreed a "just peace".
"The sooner Russia agrees to restore a just peace... the sooner the raids by the Ukrainian defence forces into Russia will stop," Ukrainian foreign ministry spokesman Georgiy Tykhy said.
Joe Biden on Tuesday said the incursion was giving Russian President Vladimir Putin a "real dilemma," in the US leader's first comments on Kyiv's surprise attack.
Since launching its invasion in February 2022, Russia has captured territory in southern and eastern Ukraine and subjected Ukrainian cities to missile and drone barrages.
- 'It's very worrying' -
More than 120,000 Russians have fled their homes in border areas of the Kursk region.
Footage from the Ukrainian TSN news outlet purportedly taken inside the Kursk region town of Sudzha showed Ukrainian soldiers climbing atop a building and removing a Russian flag, shouting "Glory to Ukraine!".
Russians in Moscow told AFP they were concerned by Ukraine's operation, which caught the Kremlin off guard.
"I have relatives living there and they refuse to leave. It's really hard," said salesperson Yulia Rusakova.
"This whole situation is a big blow. It's very hard to lead a normal, calm life, knowing that such things are happening there," she said.
Olga Raznoglazova, a 36-year-old account manager visiting from the Kursk region, said she felt the operation had brought the war closer to home.
"Now, when it is happening right next door to us... it is a completely different feeling," she said.
"It's very worrying."
Putin has vowed to "dislodge" Ukrainian troops from Russian territory, accusing Ukraine of using the operation to "improve its negotiating position" for any future talks with Moscow.
A Ukrainian security official told AFP, on condition of anonymity, that Ukraine sought to "stretch the positions of the enemy, to inflict maximum losses and to destabilise the situation in Russia as they are unable to protect their own border".
- Blindfolded POWs -
On the Ukraine side of a border crossing into Kursk, AFP reporters saw toppled concrete fortifications and caved-in remains of security and customs buildings revealing the intensity of the fighting that swept through the area.
On the road, around 10 blindfolded and bound men in Russian military fatigues were being driven in a military vehicle away from the border crossing in the direction of the city of Sumy on Tuesday.
The Russians "didn't protect the border," a Ukrainian serviceman who took part in the offensive and identified himself as Ruzhyk told AFP in Sumy region.
"They only had anti-personnel mines scattered around trees at the side of the road and a few mines that they managed to quickly throw along the highways," he said.
Another serviceman said that his unit of military engineers went in to clear the mines before Ukrainian tanks entered.
Separately, Ukrainian forces fired long-range drones at the Voronezh, Kursk, Savasleyka and Borisoglebsk airfields in central and western Russia overnight, a source in Kyiv's security services told AFP Wednesday.
The operation was the "largest attack on Russian military airfields in the entire war" and aimed at "preventing the enemy from using these airfields to launch strikes", the source said.
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