Well you see your problem here is this thing we call snowboarding. It's the act of riding a snowboard down a snow covered mountain that has unmarked obstacles on it. Sometimes these obstacles cause damage to the snowboard.
Personally I would be more worried about that grouting on the tile. That doesn't look too healthy.
Take a straight razor and just slide it across the base cutting off the dangler. Wax, snowboard, repeat as necessary.