We report on the X-ray properties of the new transient Swift J0840.7−3516, discovered with Swift/BAT in 2020February, using extensive data from Swift, MAXI, NICER, and NuSTAR. The source flux increased for∼103safter the discovery, decayed rapidly over∼5 orders of magnitude in five days, and then remained almost constant over nine months. Large-amplitude short-term variations on timescales of 1–104s were observed throughout the decay. In the initial flux rise, the source showed a hard power-law-shaped spectrum with a photon index of∼1.0extending up to∼30 keV, above which an exponential cutoff was present. The photon index increased in the following rapid decay and became∼2 at the end of the decay. A spectral absorption feature at 3–4 keV was detected in the decay. It is not straightforward to explain all the observed properties by any known class of X-ray sources. We discuss the possible nature of the source, including a Galactic low-mass X-ray binary with multiple extreme properties and a tidal disruption event by a supermassive black hole or a Galactic neutron star.