  The fully relativistic theory of the Zeeman splitting of the $(1s)^2 2s$
hyperfine-structure levels in lithiumlike ions with $Z=6 - 32$ is considered
for the magnetic field magnitude in the range from 1 to 10 T. The second-order
corrections to the Breit -- Rabi formula are calculated and discussed including
the one-electron contributions as well as the interelectronic-interaction
effects of order 1/Z. The 1/Z corrections are evaluated within a rigorous QED
approach. These corrections are combined with other
interelectronic-interaction, QED, nuclear recoil, and nuclear size corrections
to obtain high-precision theoretical values for the Zeeman splitting in Li-like
ions with nonzero nuclear spin. The results can be used for a precise
determination of nuclear magnetic moments from $g$-factor experiments.
