  Following a field-theoretical approach, we study the scalar Casimir effect
upon a perfectly conducting cylindrical shell in the presence of spontaneous
Lorentz symmetry breaking. The scalar field is modeled by a Lorentz-breaking
extension of the theory for a real scalar quantum field in the bulk regions.
The corresponding Green's functions satisfying Dirichlet boundary conditions on
the cylindrical shell are derived explicitly. We express the Casimir pressure
(i.e. the vacuum expectation value of the normal-normal component of the
stress-energy tensor) as a suitable second-order differential operator acting
on the corresponding Green's functions at coincident arguments. The divergences
are regulated by making use of zeta function techniques, and our results are
successfully compared with the Lorentz invariant case. Numerical calculations
are carried out for the Casimir pressure as a function of the Lorentz-violating
coefficient, and an approximate analytical expression for the force is
presented as well. It turns out that the Casimir pressure strongly depends on
the Lorentz-violating coefficient and it tends to diminish the force.
