Russia prides itself on its digital advances — from an extensive e-commerce services sector to ambitious plans for developing artificial intelligence.

But now, some of the key data centers at the heart of its technical prowess have been knocked out by Ukrainian drone attacks in the 4 1/2-year-old full-scale invasion, suffering damage that could take a long time to repair.

Ukraine this month hit three main data centers belonging to tech giant Yandex, one for Russia’s biggest bank, Sberbank, and one for social media platform VKontakte, the country’s equivalent to Facebook.

While Yandex, Sberbank and VKontakte are coy about the damage, people and businesses across Russia have reported widespread interruptions to cloud services.…