Spread the banana chunks out on a tray and transfer to the freezer. Leave the banana in the freezer until frozen all the way through (about one hour).
Place the frozen banana chunks into a food processor. Add the vanilla, sugar, honey or sweetener and half of the buttermilk (or yoghurt). Turn on the processor and blend for a few moments. Then, while the processor is still running, pour in the remaining buttermilk in a thin, steady stream (if you are using yoghurt, just tip it all in and blend).
Blend until the mixture is smooth and creamy. Serve at once.
The riper the bananas you use, the sweeter the flavour of ice cream.