Coming from Wiltshire, we have many examples. High Post, Malborough (the old bit), Bath, Kingsdown, Lansdown, North Wilts, South Wilts, West Wilts, Tidworth, Ogbourne. I would also suggest Painswick falls in that category.
As has been suggested, chalk or limestone is the key. This will result in a high pH and therefore like acid soils will tend to lock up nutrients and therefore inhibit growth. It is therefore ideal for golf in that a) it means not so much mowing (critical in those days), b) it is perfect for fine grass growth particularly fescue which love high ph conditions and c) the land was cheap because with all that stone about, it was near impossible for farming.
I would also add that a prerequisite for golf to be downland in variety is paradoxically that it is played on high ground, where the added wind exposure accentuates the all above and limits tree growth.
One of the great benefits of a Chalk soil is that it is very free draining but retains much water even in the driest of summers so good winter and summer conditions are usually found.