Long-lasting emission of short gamma-ray bursts (GRBs) is crucial to reveal the physical origin of the central engine as well as to detect electromagnetic (EM) counterparts to gravitational waves (GWs) from neutron star binary mergers. We investigate 65 X-ray light curves of short GRBs, which is six times more than previous studies, by combining both Swift/BAT and XRT data. The light curves are found to consist of two distinct components at $>5sigma$ with bimodal distributions of luminosity and duration, i.e., extended (with timescale $lesssim10^3$ s) and plateau emission (with timescale $gtrsim10^3$ s), which are likely the central engine activities but not afterglows. The extended emission has an isotropic energy comparable to the prompt emission, while the plateau emission has $sim0.01-1$ times of that energy. A half (50%) of our sample has both components, while the other half is consistent with having both components. This leads us to conjecture that almost all short GRBs have both the extended and plateau emission. The long-lasting emission can be explained by the jets from black holes with fallback ejecta, and could power macronovae (or kilonovae) like GRB 130603B and GRB 160821B. Based on the observed properties, we quantify the detectability of EM counterparts to GWs, including the plateau emission scattered to the off-axis angle, with CALET/HXM, INTEGRAL/SPI-ACS, Fermi/GBM, MAXI/GSC, Swift/BAT, XRT, future ISS-Lobster/WFI, Einstein Probe/WXT, and eROSITA.
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