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We present the discovery of TOI 540 b, a hot planet slightly smaller than Earth orbiting the low-mass star 2MASS J05051443-4756154. The planet has an orbital period of $P = 1.239149$ days ($\pm$ 170 ms) and a radius of $r = 0.903 \pm 0.052 R_{\rm Earth}$, and is likely terrestrial based on the observed mass-radius distribution of small exoplanets at similar insolations. The star is 14.008 pc away and we estimate its mass and radius to be $M = 0.159 \pm 0.014 M_{\rm Sun}$ and $R = 0.1895 \pm 0.0079 R_{\rm Sun}$, respectively. The star is distinctive in its very short rotational period of $P_{\rm rot} = 17.4264 +/- 0.0094$ hours and correspondingly small Rossby number of 0.007 as well as its high X-ray-to-bolometric luminosity ratio of $L_X / L_{\rm bol} = 0.0028$ based on a serendipitous XMM-Newton detection during a slew operation. This is consistent with the X-ray emission being observed at a maximum value of $L_X / L_{\rm bol} \simeq 10^{-3}$ as predicted for the most rapidly rotating M dwarfs. TOI 540 b may be an alluring target to study atmospheric erosion due to the strong stellar X-ray emission. It is also among the most accessible targets for transmission and emission spectroscopy and eclipse photometry with JWST, and may permit Doppler tomography with high-resolution spectroscopy during transit. This discovery is based on precise photometric data from TESS and ground-based follow-up observations by the MEarth team.