Are there intractable problems? Obviously, there is no problem in NP which can be shown to be intractable (by which I mean: not in P).
Is there a problem (outside of NP) which can be shown to be intractable (not lying in P)?

What you want is the Time Hierarchy Theorem, which broadly speaking states that for any "reasonable" function $f$ there are problems that take about $f(n)$ time to decide.
Concretely, each of the known EXPTIME-complete problems would be an example of something that has been proved not to lie in P.
Here is a related question on compsci.
