Is it permissible to pray on a prayer mat that covers only the hands, forehead, and nose, or must it cover the entire body?

What the woman told you is incorrect, and she must fear Allah and refrain from issuing fatwas without knowledge. It is not obligatory for someone praying on a mat or prayer rug to prostrate on it with all their limbs. The Companions prayed with the Prophet, peace and blessings be upon him, in intense heat, and they would spread out their garments and prostrate on them when they were unable to place their faces directly on the ground.