  The paper is devoted to the $\bar{K}NNN$ system, which is an exotic system
consisting of an antikaon and three nucleons. Dynamically exact four-body
Faddeev-type equations were solved and characteristics of the quasi-bound state
in the system were evaluated. Three antikaon-nucleon and three nucleon-nucleon
potentials were used, so the dependence of the four-body pole positions on the
two-body interaction models was studied. The resulting binding energies $B^{\rm
Chiral}_{K^-ppn} \sim 30.5 - 34.5$ MeV obtained with chirally motivated and
$B^{\rm SIDD}_{K^-ppn} \sim 46.4 - 52.0$ MeV obtained with phenomenological
antikaon-nucleon potentials are close to those obtained for the $K^- pp$ system
with the same $\bar{K}N$ and $NN$ potentials, while the four-body widths
$\Gamma_{K^- ppn} \sim 38.2 - 50.9$ MeV are smaller.
