  We present a comprehensive study of the critical current densities and the
superconducting vortex phase diagram in the stoichiometric superconductor
CaKFe$_4$As$_4$ which has a critical temperature of 35 K. We performed detailed
magnetization measurements both of high quality single crystals for different
orientations in an applied magnetic field up to 16 T and for a powder sample.
We find an extremely large critical current density, Jc, up to 10$^8$ A/cm2 for
single crystals when H||(ab) at 5 K, which remains robust in fields up to 16 T,
being the largest of any other iron-based superconductor. The critical current
density is reduced by a factor 10 in single crystals when H||c at 5 K and
significantly suppressed by the presence of grain boundaries in the powder
sample. We also observe the presence of the fishtail effect in the magnetic
hysteresis loops of single crystals when H||c. The flux pinning force density
and the pinning parameters suggest that the large critical current could be
linked to the existence of point core and surface pinning. Based on the vortex
phase diagram and the large critical current densities, CaKFe$_4$As$_4$ is now
established as a potential iron-based superconductor candidate for practical
applications.
