Permanent marker is only permanent in the sense that you can’t remove it with just soap/detergent and water. You can get permanent marker off most items, especially hard, non-porous items, if you use the proper products. Unfortunately, as a doll’s face is typically made of plastic or rubber (which are both semi-porous materials) it might take more time and effort to get this stain off the doll than off a harder, less porous item. Persistence should pay off, however.
First, dab a cotton swab in rubbing alcohol (or denatured alcohol) or spray the permanent marker ink directly with WD-40. These solvents help re-animate the marker ink so it is easier to remove from the surface.
Rub the cotton swab over the stain. Switch to a fresh swab as soon as the first one is dirty.
Apply more solvent and keep rubbing and dabbing until the stain is removed. Because of the porous surface, it may take a while to get all the permanent marker stain up.
Dampen a cloth and add a drop of soap or detergent to the cloth. Clean the area to remove traces of the alcohol or WD-40–these solvent substances can harm the plastic or rubber if you leave them on the doll‘s face.
You’ll find rubbing alcohol in the pharmacy and first aid sections of grocery stores and you’ll find denatured alcohol and WD-40 in the hardware sections.
You can also try nail polish remover–but it may be more damaging to the rubber or plastic than the other products listed here. If you use nail polish remover, make sure to rinse it off quickly after you’ve removed the marker stain.
Remember: Rubbing alcohol, denatured alcohol and WD-40 are all flammable–take precautions to avoid accidental fire.