It will but much more slowly. Think of cooling the blood as using the surface area of the arteries and veins as a heat sink to cool the body. Without that blood moving your next best heat sink is the skin so start piling on the ice.
The body itself is the heat sink. The skin is a radiator. The blood is the liquid in a liquid cooling system (moving heat from the sink to the radiator). To your point, liquid cooling systems become _much_ less effective when the liquid stops pumping.
