I wonder if the notion that surfing requires a surfboard is especially BE. In AE one can surf or surf the waves with no equipment. That's called body surfing.surf
• noun the mass or line of foam formed by waves breaking on a seashore or reef.
• verb 1 stand or lie on a surfboard and ride on the crest of a wave towards the shore. 2 occupy oneself by moving from site to site on (the Internet).
— ORIGIN apparently from obsolete suff, of unknown origin.