The Ukrainian military said 28 missiles were fired in the barrage.
In Lutsk, in the north-western region of Volyn, less than 100km from the Polish border, three were killed and several injured after an industrial business was hit, said regional governor Yuriy Pohulyaiko.
“As of now, three people have died. Several more wounded are in the hospital,” he wrote on the Telegram messaging app.
The building belongs to the Swedish bearing maker SKF, and the three dead were all employees at the factory, the company said.
“Last night there was an attack on the city of Lutsk in Ukraine and our factory has been hit in that attack,” SKF spokesman Carl Bjernstam said.
“We are very sad to confirm that three of our colleagues have been killed,” he added.
Mr Bjernstam said the company would look at the consequences of the attack and damage to the factory, but added that its primary focus was on its employees and their safety.
The world's biggest maker of industrial bearings has around 1,100 employees in Ukraine, the majority of whom work at its Lutsk factory, according to the company's latest earnings report.
The report also stated the factory in Lutsk accounted for around 0.5 per cent of SKF's total production volumes in 2022.
Russia said it had struck “military-purpose industrial facilities” in several sites across Ukraine.
“Ukraine's military-industrial complex suffered significant damage,” the Defence Ministry said, adding that it had carried out long-range precision strikes on “key” facilities.
The Lviv region, which is far from the front line and borders Poland to its west, was also targeted in the barrage.
No casualties were reported in Lviv but the strikes damaged more than 100 houses, broke 500 windows and destroyed a children’s playground, said officials.
“Many missiles were shot down but there were also hits in Lviv,” city mayor Andriy Sadovyi said on Telegram, adding that orders were given to evacuate at least one burning apartment building.
“There is a fire on the upper floors. We are evacuating people. All services are on site,” he said.
Mr Sadovyi posted a video of a crater outside a multistorey building with all its windows blasted out and debris from what appeared to be a playground.
At least one person was injured in a strike on Ukraine's south-eastern Dnipropetrovsk region, where a business was hit and a fire broke out, governor Serhiy Lysak said.
The Ukrainian Air Force said it had destroyed 16 of at least 28 air and sea-based missiles launched by Russia.
Earlier, Lviv regional governor Maksym Kozytskyy said “groups of Russian missiles” were headed towards the region.
Lviv region had been mostly spared Russian air attacks until this summer.
But last month seven people were killed when a missile struck a residential building.
Ukrainian media reported that, according to preliminary information, Tuesday's attack was the largest air assault on the Lviv region since the start of Russia's invasion of Ukraine in February 2022.
The city has generally been seen as a safe haven from the conflict, with some government offices moving there and international NGOs using it as a base. It has also been a transit point for Ukrainian refugees en route to Poland and beyond.
“These are the parts of the country where millions of people are seeking safety and refuge after fleeing the horrors of Russia's invasion,” Denise Brown, the UN's resident co-ordinator in Ukraine, said in a statement.
“Russia's persistent attacks hitting essential infrastructure in populated areas cause immense human suffering.”
Air raid alerts were issued for all of Ukraine for about two hours, from about 2am local time.
“The daily terror of the Russians has a single goal: to break us, our spirit for fighting,” Andriy Yermak, the head of Ukraine's presidential administration, wrote on Telegram.
“This will not happen.”
Meanwhile, Russia's Defence Minister Sergey Shoigu claimed on Tuesday that Ukraine's ability to fight had been “almost exhausted” and said the war had exposed vulnerabilities in western weapons systems that Moscow would soon share.
Speaking at a security conference in Moscow attended by Chinese Defence Minister Li Shangfu, Mr Shoigu said the conflict had been a serious test for Russia.
“In the special military operation, the Russian army has debunked many myths about the superiority of Western military standards,” Mr Shoigu said in a rare public speech.
“The preliminary results of combat operations show that Ukraine's military resources are almost exhausted.”
Ukraine and Russia conflict latest – in pictures
Separately, Russia fined social media site Reddit on Tuesday for the first time for not deleting “banned content” that it said contained “fake” information about Russia's military campaign in Ukraine, state news agency RIA reported on Tuesday, citing a Moscow court.
Reddit joins a list of sites under scrutiny in Russia for failing to remove content that Moscow deems illegal, including Wikipedia, streaming service Twitch, and Google.
RIA said the court had fined Reddit 2 million roubles ($20,365). Reddit did not immediately respond to a request for comment.
Also on Tuesday, the court fined the Wikimedia Foundation, which owns Wikipedia, 2 million roubles, for failing to delete “fakes” about what Moscow calls its “special military operation” in Ukraine, Interfax reported.