My two "sports" are chess and stairclimbing. I use quotes over sports because some don't see them as much of a sport.
In chess, mindfulness is obviously very important.
"A weaker player with better concentration will out-perform a more intelligent player with less concentration" -- Mencius (Confucian scholar).
In stairclimbing, the stairwells are usually dark and boring and there is plenty of opportunity for mindfulness just to avoid the monotony of climbing with no views, not much to please the senses, no temptations, etc. But one of the best parts about stairclimbing is that you get done fast. A 10 minute workout with stairs is equivalent to about a 20 minute run (because of moving your body weight vertically, against gravity, instead of horizontally).