Permutation for which an inverse is a hard problem
It is obviously possible; one simple example is the function $f(x) = g^x - 1 \bmod p$, where $p$ is a large prime, and $g$ generates the entire group $\mathbb{Z}_p^*$; this is permutation over the range $[0, p-1)$, and is difficult to invert if the discrete log problem is hard.

What may be more difficult is to design an efficient hard-to-invert permutation; most efficient permutations are designed as a composition of trivially invertible permutations (and hence the result can be easily inverted).