  For positive integers $m$ and $k$, we introduce a family of lattices
$\mathcal{C}_{k}^{(m)}$ associated to the Cambrian lattice $\mathcal{C}_{k}$ of
the dihedral group $I_{2}(k)$. We show that $\mathcal{C}_{k}^{(m)}$ satisfies
some basic properties of a Fuss-Catalan generalization of $\mathcal{C}_{k}$,
namely that $\mathcal{C}_{k}^{(1)}=\mathcal{C}_{k}$ and
$\bigl\lvert\mathcal{C}_{k}^{(m)}\bigr\rvert=\mbox{Cat}^{(m)}\bigl(I_{2}(k)\bigr)$.
Subsequently, we prove some structural and topological properties of these
lattices---namely that they are trim and EL-shellable---which were known for
$\mathcal{C}_{k}$ before. Remarkably, our construction coincides in the case
$k=3$ with the $m$-Tamari lattice of parameter 3 due to Bergeron and
Pr{\'e}ville-Ratelle. Eventually, we investigate this construction in the
context of other Coxeter groups, in particular we conjecture that the lattice
completion of the analogous construction for the symmetric group
$\mathfrak{S}_{n}$ and the long cycle $(1\;2\;\ldots\;n)$ is isomorphic to the
$m$-Tamari lattice of parameter $n$.
