Most definitely, yes you can cool your whole body by cooling only a small region for a very clear reason: the human vascular system (the same can be said for any animal with a heart-pumped circulatory system).
Heat flows through conduction out of the blood in the locally cooled region into whatever is cooling it - the ice pack, or the skin cooled evaporatively / convectively. Then this slightly cooled blood is pumped to all other parts of the body; heat can then be conducted from other body parts into the blood. The cycle repeats, so the blood is constantly ferrying heat through conduction from throughout the body to the locally cooled region.
Indeed, this principle is also widely used in industry for the disposal of waste heat or transfer of heat to where it is needed. Coolant is often cooled by cooling towers and then pumped through the industrial plant whence we want to dispose of heat - the heat to be disposed of is generated remotely from the cooling tower. Likewise, a liquid sodium "vascular system" is used to transfer heat from the core of a nuclear reactor to the remote heat exchanger, where it is cooled through the production of steam for turbines and the attendant loss heat to supply the latent vaporization heat for the water.