We present a new search for hypothetical massive, charged, long-lived particles with the ATLAS detector at the LHC using an integrated luminosity of 140 fb$^{-1}$ of proton-proton collisions at $\sqrt{s}$=13 TeV. These particles are expected to move significantly slower than the speed of light and should be identifiable by their high transverse momenta and anomalously large specific ionisation losses measured in the pixel detector. This information can be used in combination with the speed measured by Time-of-Flight in the ATLAS calorimeters. Results are presented covering particles with lifetimes down to O(3) ns and with masses ranging from 200 GeV to 3 TeV. Interpretations for pair-production of R-hadrons and staus in scenarios of supersymmetry compatible with these particles being long-lived are presented, with mass limits extending beyond those from previous searches in broad ranges of lifetime.