  We analyze the thermodynamic Casimir effect occurring in a gas of
non-interacting bosons confined by two parallel walls with a strongly
anisotropic dispersion inherited from an underlying lattice. In the direction
perpendicular to the confining walls the standard quadratic dispersion is
replaced by the term $|{\bf p}|^{\alpha}$ with $\alpha \geq 2$ treated as a
parameter. We derive a closed, analytical expression for the Casimir force
depending on the dimensionality $d$ and the exponent $\alpha$, and analyze it
for thermodynamic states in which the Bose-Einstein condensate is present. For
$\alpha\in\{4,6,8,\dots\}$ the exponent governing the decay of the Casimir
force with increasing distance between the walls becomes modified and the
Casimir amplitude $\Delta_{\alpha}(d)$ exhibits oscillations of sign as a
function of $d$. Otherwise we find that $\Delta_{\alpha}(d)$ features
singularities when viewed as a function of $d$ and $\alpha$. Recovering the
known previous results for the isotropic limit $\alpha=2$ turns out to occur
via a cancellation of singular terms.
