The history of prone surfing or "bellyboarding" dates back to the 1920's in the UK. The plywood body board is a descendent of the Hawaiian paipo board. They were made by shapers like Devon's DIck Pearce, who made boards for over 50 years and perfected the 'surf rider' template. These wooden style of boards started to disappear as polystyrene foam boards became popular in the 1970's, but have recently had a resurgence thanks to events such as the Bellyboard World Championships which is held in St Agnes, Cornwall every year in September.
For more info on the history of prone and stand up surfing, check out the amazing Museum of British Surfing in Braunton, Devon.