Yes, winter activities are possible in Kenya. Mount Kenya is the second highest mountain in Africa and reaches high enough to have enough snow to ski on.
If you are an avid skier or snowboarder, you will likely want to avail yourself of the opportunity of skiing/boarding so close to the Equator (mind-boggling indeed, and not many people can say that they have). Elephant, rhino, buffalo and leopard roam the surrounding areas so you could have the occasional sighting as you go downhill. Truly a unique experience. Unfortunately with global warming, this might soon be a thing of the past as the glaciers continue to melt away.
Ice climbing is also really popular and the South Face route on Batian is considered a serious climb for experienced rock climbers and alpinists. According to literature about it, and the guides who lead the climbs, excellent physical condition is a must.
There are several different companies that offer skiing and ice climbing expeditions. You can also visit the Mount Kenya National Park official website for suggestions.