Neoprene bucket hat with liner. One size cut to fit 54cm and stretchable. Pictures show black neoprene bucket hat with "zebra surf" liner.
This hat stays on because it stretches to fit and it keeps the sun off. It is quite warm as it's made of nylon-lined 3mm neoprene (polychloroprene), so it can be hot in the summer.
This hat is completely waterproof. Seams under liner are sewn, glued and taped, and liner is stretchable fabric with DWR (Durable Water Repellent) finish.
