The article describes an experimental study, which aims to detect signs of jump-like inelasticity in sandstone and to establish their influence on the P-wave attenuation. The measurements were taken in a sandstone sample using the reflection method at a pulse frequency of 1 MHz and with the strain amplitudes 1 0.1 microstrain and 2 0.24 microstrain at a hydrostatic pressure of 10 MPa. Using the high-resolution recording of the acoustic signal, the short-period low-amplitude oscillations were detected in the waveform and were interpreted as manifestations of jump-like inelasticity. These unconventional manifestations have forms of a stress plateau and a stress drop. At all strain amplitudes, with an increasing frequency in the range of (0.52.0) MHz, the P-wave attenuation in the sandstone sample monotonously increases and has a peak at a frequency of 1.56 MHz. The jump-like process is amplitude-dependent; it introduces changes in the attenuation peak parameters. The simple simulation demonstrated considerable variation in the attenuation peak magnitude (from 0.353 to 0.607) due to this effect. The jump-like deformation process in rocks can presumably be explained by the combined action of the viscoelastic and microplastic mechanisms. The results of this study can be useful for improving the methods of geological interpretation of seismic and acoustic data.