No, you can't make ice cream without dairy products. But you CAN make some very tasty frozen desserts that will give you a similar mouthfeel to ice cream. Several really good ideas are already in this thread.
I agree with ChefJune. Ice cream is an egg and dairy based custard. You can try to replicate it (God knows why) but it's no longer ice cream. If you are happy with the outcome as a dessert, then lucky you. Same Devil's Bargain with "frozen yogurt".
Theres a recipe with frozen bananas here at Food52 from Genius Recipes...Cut the bananas in small pieces,then you let them thaw a bit and blend in the food processor till the texture is right.You can add chocolate or nuts or some other fruits to make other flavours.You have to do this at serving time,but there will be no waste cause you can freeze and blend it again.
http://food52.com/recipes/18742-the-kitchn-s-one-ingredient-ice-cream http://food52.com/recipes/7418-vegan-pumpkin-pie-ice-cream here are 2 ways you can make it
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And one using bananas, http://www.keepingittasty.com/2013/01/22/peanut-butter-banana-ice-cream-with-chocolate-chunks-and-no-churn-needed/