Ukrainian troops have retaken seven villages spanning 35 square miles from Russian forces in the past week, the deputy defense minister said, as the early stages of Kyiv’s counteroffensive notched incremental successes.

That comes as encouraging news following reports that Russian missiles hit civilian buildings in the central Ukrainian city of Kryvyi Rih overnight, killing at least 10 people and wounding more than two dozen.

Deputy Defense Minister Hanna Maliar wrote on the Telegram app that the Ukrainian flag was again flying over the village of Storozhov, in the eastern Donetsk province, and that her troops had also retaken three other nearby small villages and three in neighboring Zaporizhzhia province.

“The battles are tough, but our movement is there, and that is very important,” President Zelensky said in his nightly video address. He added that rainy weather is challenging his troops, and that he’s discussed with his military commanders “which points of the front we need to strengthen and what actions we can take to break more Russian positions.”

On Sunday, Ukrainian officials said their troops took the Donestk villages of Blahodatne, Makarivka, and Neskuchne — south of the town of Velyka Novosilka. Ms. Maliar reported Monday that the Zaporizhzhia province settlements of Lobkove, Levadne, and Novodrivka were also now back under Ukrainian control.

Russian officials did not confirm Ukraine’s gains, which were impossible to verify and could be reversed in the to-and-fro of war. The gains amounted to only small bits of territory and underscored the difficulty of the battle ahead for Ukrainian forces, who will have to fight meter by meter to regain the roughly one-fifth of their country under Russian occupation.

Recent fighting on the western edge of the 600-mile front line has been complicated by a dam breach that sent floodwaters into a part of the Dnieper River separating the two sides.

Western analysts and military officials have cautioned that an effort to rid Ukraine of entrenched and powerfully armed Russian troops could take years, and the success of the Ukrainian counteroffensive is far from certain. President Macron said at Paris that the Ukrainian counteroffensive began several days ago and “is set to be deployed over several weeks, if not months.”

An official with the Moscow-appointed administration of the Zaporizhzhia region at the western end of the front line, Vladimir Rogov, said “heavy battles” were raging in the area Monday involving Russian artillery, mortars and air power.

The villages are part of an area where the Russian front lines jut out into territory held by Ukraine. While just more than a mile deep, the protrusion has recently become one of several epicenters of intense fighting along the front line that cuts across southern and eastern Ukraine.

Despite the villages’ small size, their capture involved an incursion into the first line of Russian defenses and could allow Ukrainian forces to try a deeper thrust into occupied areas.

Russian forces control far less Ukrainian land than they did before a blistering Ukrainian counteroffensive last year that retook the northern city of Kharkiv and southern city of Kherson, among other places. 

On Saturday, Mr. Zelensky said “counteroffensive, defensive actions are taking place,” without specifying whether it was the all-out counteroffensive that has long been expected after a vast infusion of Western firepower and air defense systems into Ukraine. A day earlier, President Putin asserted that the counteroffensive had started and Ukrainian forces were taking “significant losses.”

Ukrainian forces have focused on the Zaporizhzhia region and an area near the devastated Donetsk city of Bakhmut, among other locations. Russian authorities have said their troops are largely holding their ground.

Yet a prominent Russian military blogger who goes by the nickname WarGonzo, Semyon Pegov, acknowledged Russian troops had withdrawn from Blahodatne, Neskuchne, and Makarivka, and said Ukrainian forces were trying to push forward along the banks of the Mokri Yaly River on Monday.

A military correspondent for the Russian daily Komsomolskaya Pravda, Alexandet Kots, said Ukrainian forces were attempting to advance, despite heavy losses, toward the town of Staromlinovka, which sits on a strategic highway leading to the port city of Mariupol. Russian forces captured the city more than a year ago, after Ukrainian forces held out for several months in a grueling and desperate defense.

Separately Monday, the Ukrainian Defense Ministry said at least four civilians were killed and 16 others wounded by Russian shelling over the last 24 hours.

In Donetsk, Russian shelling hit nine towns and villages and left one civilian dead and two others wounded. The Donetsk governor, Pavlo Kyrylenko, posted images of apartment buildings and a cultural center damaged by Russian strikes in the town of Avdiivka.

In Kharkiv, to the north, Russian forces pummeled several settlements with artillery and mortar and rocket fire, wounding at least three people, the regional state administration chief, Oleh Synehubov, wrote on Telegram.

The reported Ukrainian advance came as authorities on both sides of the front line pressed on with rescue and relocation efforts for civilians in the Kherson region driven from their homes by flooding from the breach of the Kakhovka dam last week.

On Sunday, Mr. Prokudin said three people were killed when Moscow’s troops opened fire on a boat evacuating people from Russian-occupied areas toward Ukrainian-held ones.

As for the Russian missile attack on Kryvyi Rih — Volodymyr Zelensky’s hometown — images from the scene relayed by Mr. Zelensky on his Telegram channel showed firefighters battling a blaze as pockets of fire poked through multiple broken windows of a building. Charred and damaged vehicles littered the nearby ground.

“More terrorist missiles,” Mr. Zelensky wrote. “Russian killers continue their war against residential buildings, ordinary cities and people.”