If you are using the right clay mix (or clay bricks), moisture and tamping, you should be able to create a base that is very difficult to dig too far into. Frequent re-application is necessary.
Regarding your idea of rubber mats, there are actually products sold that are, I believe, rubber that are designed for just what you are talking about. It is not solid, though. I think it is like a honeycomb grid configuration where dirt fills in the openings throughout. I couldn't find on a quick search but maybe someone else will chime in. Also, Turface.com is a great resource.