  We have studied the existence of topological self-dual configurations in a
nonminimal CPT-odd and Lorentz-violating (LV) Maxwell-Higgs model, where the LV
interaction is introduced by modifying the minimal covariant derivative. The
Bogomol'nyi-Prasad-Sommerfield formalism has been implemented, revealing that
the scalar self-interaction implying self-dual equations contains a derivative
coupling. The CPT-odd self-dual equations describe electrically neutral
configurations with finite total energy proportional to the total magnetic
flux, which differ from the charged solutions of other CPT-odd and LV models
previously studied. In particular, we have investigated the axially symmetrical
self-dual vortex solutions altered by the LV parameter. For large distances,
the profiles possess general behavior similar to the vortices of
Abrikosov-Nielsen-Olesen. However, within the vortex core, the profiles of the
magnetic field and energy can differ substantially from ones of the
Maxwell-Higgs model depending if the LV parameter is negative or positive.
