Are there any computational problems in groups that are harder than P?

An earlier reference for groups with this property is

J. Avenhaus and K. Madlener. Subrekursive Komplexität der Gruppen. I. Gruppen mit vorgeschriebenen Komplexität. Acta Infomat., 9 (1): 87-104, 1977/78.

There is a hierarchy of the recursive functions known as the (difficult to pronounce) Grzegorczyk Hierarchy $E_0 \subset E_1 \subset E_2 \subset \cdots$, where (roughly) $E_1$ contains the linearly bounded functions, $E_2$ polynomially bounded functions, and $E_3$ those functions that are bounded by iterated exponentials.

The above paper describes constructions of finitely presented groups $G_n$ for $n \ge 3$, in which solving the word problem has time complexity bounded by a function $E_n$ but not by any function in $E_{n-1}$,

As Andreas says (in his answer and his comment to it), there are groups whose word problem is undecidable and one could similarly set up a group that encodes the halting problem of a class of Turing machines where this is decidable but difficult. However, one must be careful in the encoding. In

Isoperimetric and Isodiametric Functions of Groups, Mark V. Sapir, Jean-Camille Birget and Eliyahu Rips Annals of Mathematics Second Series, Vol. 156, No. 2 (Sep., 2002), pp. 345-466

and

Mark V. Sapir, Jean-Camille Birget and Eliyahu Rips Isoperimetric functions of groups and computational complexity of the word problem. Ann. of Math. (2) 156 (2002), no. 2, 467–518.

groups with NP complete word problem are constructed and other similar results. See in particular, Corollary 1.1 of the first paper listed above.

To add more information, Corollary 1.1 says:

There exists a finitely presented group with NP-complete word problem. Moreover for every language $L\subseteq A^*$ for some finite alphabet $A$ there exists a finitely presented group $G$ such that the nondeterministic complexity of $G$ is polynomially equivalent to the nondeterministic complexity of $L$.

So, for instance, if $L$ is an EXP-time complete problem, then the word problem of $G$ is in NEXP-time and not in NP (hence not in P). Of course you can replace EXP by your favorite time complexity class strictly above NP.

There are finitely presented groups whose word problem is undecidable. See, for example, https://en.wikipedia.org/wiki/Word_problem_for_groups .