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Its never happened to me. I've probably ridden through -20C half a dosen times. So cant say for sure. You get into the habit of moving stuff a lot though. Changing gear back and forth, pumping brakes a bit etc.
Also staying clear of salted roads is a must below -15C. I nearly snapped a leg after road slush refroze, Welding my feet to my pedals. Luckily the base of my shoe snapped instead.
Until hot brakes melt the snow that's built up on them. I suppose it's unlikely, and that crossers in -10C are more likely to get their brakes hot enough to melt the snow than fatbikers trundling across the ice caps.