In structural and biomedical applications, where high-speed loading is a major concern for mechanical reliability, high-performance polymers (HPP) are increasingly required. Still, their thermomechanical response produced by additive manufacturing methods has not been fully characterized. In this study, the strain-rate-dependent compressive behavior of specimens, 3D printed by the material extrusion (MEX) method with the polyether ether ketone (PEEK) biopolymer, has been investigated. Simultaneously, the evolution of specimen temperature has been monitored using an infrared camera. Compression testing was performed over a range of test speeds up to 200 mm/min. The aim was to quantify the mechanical performance under compression loads and concurrently thermal self-heating phenomena as a function of applied strain rate. Results show that the compressive strength of the MEX-processed PEEK had a positive strain-rate sensitivity of 14.1% at higher strain rates. The strain-rate sensitivity index had higher values at lower test velocities, suggesting that viscoelastic effects play a larger role in the deformation mechanism. At the same time, the maximum specimen temperature increased by 33% as the strain rate increased (63 to 85 °C) (thermomechanical self-heating), which can affect material response at elevated strain rates. This work offers fundamental insights into the mechanical response of MEX-fabricated PEEK biopolymer under loads applied at various speeds and has direct merit for the design of additively manufactured biomedical components which are often subjected to such loading conditions.