An attorney who is the son-in-law of one of Russia’s richest men pleaded guilty Tuesday to lying to Special Counsel Robert Mueller’s investigators as Mr. Mueller’s team continued to aggressively pursue its wide-ranging probe into Russia’s meddling in the 2016 election.

Alex van der Zwaan, a Dutch national and a former associate in London at the law firm Skadden, Arps, Slate, Meagher & Flom LLP, admitted he lied about his contact with Richard Gates, a former adviser to President Donald Trump, in a November interview about...