Classes between $\textbf{PSPACE}$ and $\textbf{EXP}$
Depends what you mean by "class". 

I don't know of any natural class that satisfies that, but certainly one can do the following:

Consider the set of languages that are decidable in time $2^n$ (exactly, not just any exponential). This is strictly contained in EXP by the time-hierarchy theorem.
Take the union of this set and PSPACE, and you have a class that strictly contains P (again, time hierarchy), contains PSPACE, and is contained in EXP (and presumably, if PSPACE$\neq$ EXP, then there is also strict containemnt here).