Kate asked about an indoor arena and THAT is a whole other issue.
Last winter we were at a different barn and Nina was ridden in a heated indoor arena nearly every day. There were lots of distractions, doors, windows to the aisle and she might spook a little when someone appeared in a doorway, but overall she was fine. It was a solid old pole barn and you really couldn't tell what the weather was doing when it was closed up tight. Here she is indoors last winter.