Ukrainian civilians grappled with a second day of extensive cellular phone and internet outages following a devastating cyberattack on Tuesday.

The attack – orchestrated by Kremlin-backed hackers – targeted Kyivstar, the country's largest mobile phone and Internet provider, affecting 24.3 million mobile subscribers and over 1.1 million home Internet users.

Two hacking groups, Killnet and Solntsepek – both linked to the Russian government – claimed responsibility...