  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.
