Looks like fun, but they invented snowboards for that

What also looks like fun is the snow bodyboards from Airboard (who also make SUPs - actually, they were making those snow bodyboards long before the SUP days):

www.airboard.com/

And I also plan to try the Bellyak this winter, which according to the manufacturer should be good fun on snow:

bellyak.com/

Yes but snowboards can't go in the water.... We went down the hill and right into the river for a paddle

Wouldn't that mess up your boards?

They only get messed up if you crash! otherwise only few scuffs