Britain’s vote to leave the European Union battered the British pound by more than 11% Thursday night and into Friday morning, pushed down stocks in Asia and pointed to a day of steep falls across the world’s financial markets. It was a brutal drubbing for investors who had stacked up bets that the U.K. would choose to stay.

The pound traded as low as $1.3230 in London’s early hours, as a pink-fringed dawn rose over the city and the British Broadcasting Corp. called the referendum for Leave. That was the lowest level against...