Truthfully, there are still faint marks after the rubbing alcohol treatment. It's a huge improvement. I will try some of these other suggestions....after the 25th. I've been ill so everything is behind schedule. I didn't want to ruin the surface of the mat or use something that would mark fabric. Thank you to all of you for your ideas.

I make a paste with baking soda and vinegar. So far it's taken marker off everything I've used it on!

You can try rubbing alcohol or acetone-based nail polish remover. Start by dabbing a little amount of rubbing alcohol into a cotton cloth to remove permanent marker from a desk. To remove the residue, lightly dab the area to moisten it. Then, using a wet towel, wipe the area clean. If a small mark remains, simply repeat the procedure.

BANG :D
Well YOU said give it a shot!

Did anyone try a Mr. Clean eraser?

I'm sorry but PLEASE understand I was awake for almost 48 hours when I posted the above. I must have been more out of it than even I knew at that point. Again I'm sorry.

This post is from seven years ago, I doubt the poster is still looking for an answer.

there was a spam post that revived it. It was removed.

Acetone will take sharpie off. I mark my machines with sharpie and it comes off every time. Also removed sharpie from the edge of rulers. Many more things.

Now I know what to do with all those bottles of hand sanitizer I bought when we were in the height of the Covid pandemic. I still use it but have about enough to last 3 years at this rate!