We report the temperature-pressure phase diagram of CaKFe4As4 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 CaKFe4As4 is suppressed and then disappears at p≳4 GPa. High-pressure x-ray data clearly indicate a phase transition to a collapsed tetragonal phase in CaKFe4As4 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 independent of pressure, occurring at 4.0(5) GPa for temperatures below 150 K. Density functional theory calculations also find a sudden transition to a collapsed tetragonal state near 4 GPa, as As-As bonding develops across the Ca layer. Bonding across the K layer only occurs for p≥12 GPa. These findings demonstrate a different type of collapsed tetragonal phase in CaKFe4As4 as compared to CaFe2As2: a half-collapsed tetragonal phase.