They do, but they can get pretty expensive. For example:
Pokemon Card Game DP Official Play Mat by Media Factory
You can order in bulk: https://custommousepad.com/index.php?route=product/product&product_id=334
There are also resources for fan created images that you could have custom printed at sites like https://custommousepad.com
http://icycatelf.deviantart.com/art/Simple-Pokemon-TCG-Playmat-Template-446733457
http://pokegym.net/forums/showthread.php?71837-Pokemon-Custom-Mats
Given how durable the paper mats are if not unduly abused, and how many constructed decks have come into the house over the last year, I don't see my wife and I buying a custom Pokemon mat anytime soon, if ever. By the time we burn through the paper mats we have, we'll all know the game well enough to just use a generic play mat to protect the cards!
From what I've seen, most people with custom Pokemon mats prefer Pokemon artwork such as a dramatic image of their favorite Pokemon to one that has the rules and card layout.
Finally, there's a Wizards of the Coast cloth playmat (no rubber backing) that had a limited run and can be found on ebay for about $25 to $30.
Given your situation, I'd recommend starting with the paper playmat that comes with constructed decks. After you're all used to the game, switch to a generic rubberized mat for actual play and hang onto the paper one (or what's left of it at that point) for reference.
Remember, you're introducing Pokemon to a 4 year old. There's a finite amount of money that it makes sense to spend on this experiment!