No. The cap is stainless steel, with one food-grade silicone ring so it doesn't leak.
Almost every steel bottle on the market closes with plastic. Hydro Flask, in their own FAQ: "All of Hydro Flask's caps are made out of BPA/BPS free polypropylene." Chilly's, Stanley and Mepal are the same. Klean Kanteen — the most credible plastic-free brand in the category — ships polypropylene caps as standard and charges extra for one that isn't.
Ours is steel as standard, and it costs less than all of them.