magnetic mat for card sorting game
It seems like what you need is not magnetic mats, but instead simply a metal playmat, and magnetic cards. Unless you produce it yourself, I doubt you're going to find magnetized sheets that fold, whereas you might find a foldable metal whiteboard, or could build one yourself. From there, making the magnetic cards could be as easy as attaching magnets to the back of paper cards, or get the cards printed on magnetic sheets. Of course, how feasible this is depends on your board and card sizes.

The combination of metal and foldable sounds like it wouldn't be much fun sliding things over the surface, nevermind if either part is magnetic. Since you're already dealing with uncomfortable movability, have you considered using hook and loop fasteners (also called Velcro, because of it's manufacturer/trademark holder) instead? This would be consideably cheaper and more portable than a solution involving metal components, nevermind less likely to break and - if you make everything out of fabric material - even washable.

If all you need is a solution to be able to fix the cards into place, you might want to go with a wooden base plate and use tacks to fix the cards on the surface once you're satisfied with the position. This is the most budget-efficient variant I can think of. It comes with the downside of not being infinitely reuseable, as both the board and the cards will be damaged a little bit each time they're used, but this is certainly something you could go with for testing purposes one way or another.

You could also go more fancy and use a base layer out of any rigid material of your choice, along with a cover layer out of a transparent, equally rigid material such as acryllic glass. You could move the cards around without the inhibiting factor a magnet would have, and when you're done, place the transparent layer on top and affix it in a suitable way that depends on what you're trying to do with it.
You may be fine with some kind of clamps, but if you need to pin it on a wall or need to otherwise store it in an upright state, you would be better off using screws (you could fix them to the base plate, reducing the amount of loose parts while making the final assembly easier) and wing nuts, which are easy to use and don't require tools.

If I'm missing what you're trying to do entirely, please clarify your question, providing details about anything you can think of. Brainstorming requires some kind of ground under the feet.