  We investigate the Casimir effect, due to the confinement of a scalar field
in a $D$-dimensional sphere, with Lorentz symmetry breaking. The
Lorentz-violating part of the theory is described by the term $\lambda (u \cdot
\partial \phi) ^{2}$, where the parameter $\lambda$ and the background vector
$u^{\mu}$ codify the breakdown of Lorentz symmetry. We compute, as a function
of $D$, the Casimir stress by using Green's function techniques for two
specific choices of the vector $u ^{\mu}$. In the timelike case, $u ^{\mu} =
(1,0,...,0)$, the Casimir stress can be factorized as the product of the
Lorentz invariant result times the factor $(1 + \lambda) ^{-1/2}$. For the
radial spacelike case, $u ^{\mu} = (0,1,0,...,0)$, we obtain an analytical
expression for the Casimir stress which nevertheless does not admit a
factorization in terms of the Lorentz invariant result. For the radial
spacelike case we find that there exists a critical value $\lambda _{c} =
\lambda _{c} (D)$ at which the Casimir stress transits from a repulsive
behavior to an attractive one for any $D> 2$. The physically relevant case $D =
3$ is analyzed in detail where the critical value $\lambda _{c}|_{\small D=3} =
0.0025$ was found. As in the Lorentz symmetric case, the force maintains the
divergent behavior at positive even integer values of $D$.
