HONG KONG—Authorities arrested an elected opposition politician under a British colonial-era anti-sedition law after she shared a Facebook post containing personal information of a police officer who allegedly shot a journalist, as recriminations continue from the protests that swept the city for more than half a year.

Around 1 a.m. Thursday, Hong Kong police arrested Cheng Lai-king, a member of the Democratic Party who serves as chairwoman of her district’s local council. Police said Ms. Cheng had been arrested for “seditious...