  We report the temperature-pressure phase diagram of CaKFe$_4$As$_4$
established using high pressure electrical resistivity, magnetization and high
energy x-ray diffraction measurements up to 6 GPa. With increasing pressure,
both resistivity and magnetization data show that the bulk superconducting
transition of CaKFe$_4$As$_4$ is suppressed and then disappears at $p$
$\gtrsim$ 4 GPa. High pressure x-ray data clearly indicate a phase transition
to a collapsed tetragonal phase in CaKFe$_4$As$_4$ under pressure that
coincides with the abrupt loss of bulk superconductivity near 4 GPa. The x-ray
data, combined with resistivity data, indicate that the collapsed tetragonal
transition line is essentially vertical, occuring at 4.0(5) GPa for
temperatures below 150 K. Band structure calculations also find a sudden
transition to a collapsed tetragonal state near 4 GPa, as As-As bonding takes
place across the Ca-layer. Bonding across the K-layer only occurs for $p$
$\geq$ 12 GPa. These findings demonstrate a new type of collapsed tetragonal
phase in CaKFe$_4$As$_4$: a half-collapsed-tetragonal phase.
