According to the Encyclopaedia Britannica, the earliest form of golf is attributed to the Romans. Over 2,000 years ago, as Rome expanded its borders as far north as Britannia, invading armies carried with them a home-grown game called paganica, practised with a stick and a stitched leather ball stuffed with feathers. The idea of hitting a ball as far as possible towards a defined target seemed to catch on. Through the centuries, games similar to paganica developed in England, France, Italy, the Netherlands and China. Modern golf developed in Scotland where the feathery ball, identical to that used for paganica, was standard for several centuries. It was replaced only in the mid-19th century when the first rubber ball – the guttie – was introduced.
The only Encyclopedia Britannica I've got here is the 1911 version and it doesn't suggest this. Searching the web for paganica has come up pretty much empty too, although there is a golf course in Wisconsin with that name. A golf course in Portugal mentions paganica in passing. The World Book seems to have this same claim (i.e. about sticks and leather balls) ... so, does anyone have an ancient source which mentions paganica (as a game?)? I'll note in passing that paganica is also brought up as the 'origin' of hockey (presumably of the field variety) ... maybe if we could establish the Roman origins of hockey, Classics might have a resurgence up here in the Great White (well, windy today) North.