ZURICH——A top executive at Swiss drugmaker Novartis AG is stepping down amid the controversy over payments that corporations made to a company owned by U.S. President Donald Trump’s longtime personal lawyer .

Novartis said Wednesday that general counsel Felix Ehrat, 61 years old, is retiring from the company in connection with $1.2 million in payments it made over the course of a year to Michael Cohen’s shell company, Essential Consultants LLC.

It...