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With the lack of moisture, that would come from the skin, could a natural fibre substance (e.g. polar bear fur) possibly stick to a surface of ice?

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Pretty much anything can stick to ice if pressed against it. This occurs because the solid-liquid transition in water is quite unusual, with the liquid form being more dense than the solid. That means that ice can be melted by pressure alone; the pressure moves the material toward its densest state, which is liquid water. This is actually what makes ice skating possible; a thin layer of liquid melts under the skate blades, which then can slide freely across this ductile liquid.

That means that if you press a patch of solid material against ice hard enough (how hard exactly depends on how cold the ice is; it's easier to pressure melt it closer to $0^{\circ}$ C), the ice can melt. The liquid released will slow out to the sides, until the pressure on it is relived, at which point it promptly refreezes. This is what is responsible for ice cubes sticking together, but it works, in principle, with just about anything you might want to press against an icy surface.

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