Do you have ditches you can ride in? They usualy don't freeze up too bad (where do you live? I don't think it gets colder than in Calgary
We board at a private ranch with no arena at all but I work my horses on the trail too, or just on the gravel road. You don't have to ride a horse hard to work it and even on hard ground you can flex them, side pass, shoulder in, small circles etc. and work on your seat at the same time...
Just my 0.02$