  The unconventional superconductivity in the noncentrosymmetric LaNiC$_2$, and
its evolution with pressure, is analyzed basing on the {\it ab initio}
computations and the full Eliashberg formalism. First principles calculations
of the electronic structure, phonons and the electron-phonon coupling are
reported in the pressure range 0-15 GPa. The thermodynamic properties of the
superconducting state were determined numerically solving the Eliashberg
equations. We found that already at $p=0$ GPa, the superconducting parameters
deviate from the BCS-type, and a large value of the Coulomb pseudopotential
$\mu^{\star}=0.22$ is required to get the critical temperature $T_c = 2.8$~K
consistent with experiment. If such $\mu^{\star}$ is used, the Eliashberg
formalism reproduces also the experimentally observed values of the
superconducting order parameter, the electronic specific heat jump at the
critical temperature, and the change of the London penetration depth with
temperature. This shows, that deviation of the above-mentioned parameters from
the BCS predictions do not prejudge on the triplet or multiple gap nature of
the superconductivity in this compound. Under the external pressure,
calculations predict continuous increase of the electron-phonon coupling
constant in the whole pressure range 0-15~GPa, consistent with the
experimentally observed increase in $T_c$ for the pressure range 0-4~GPa, but
inconsistent with the drop of $T_c$ above 4~GPa and the disappearance of the
superconductivity above 7~GPa, reported experimentally. The disappearance of
superconductivity may be accounted for by increasing the $\mu^{\star}$ to 0.36
at 7~GPa, which supports the hypothesis of the formation of a new high-pressure
electronic phase, which competes with the superconductivity.
