  TUS (Tracking Ultraviolet Set-up) was the world's first orbital detector
aimed at testing the principle of observing ultra-high energy cosmic rays
(UHECRs) with a space-based fluorescence telescope. TUS was launched into orbit
on 28th April 2016 as a part of the scientific payload of the Lomonosov
satellite, and its mission continued for 1.5 years. During this time, its
exposure reached $\sim1550$ km$^2$ sr yr for primary energy $\gtrsim400$ EeV,
and a number of extensive air showers-like events were registered. The shape
and kinematics of the signal in these events closely resembled those expected
from UHECRs but amplitudes of the signal and some other features were in
contradiction with this assumption. A detailed analysis of one of EAS-like
events (TUS161003) revealed that a primary cosmic ray would need to have an
energy $\gtrsim1$ ZeV in order to produce a light curve of the observed
amplitude, which is incompatible with the cosmic ray spectrum obtained with
ground-based experiments. More than this, the slant depth of the shower maximum
be the signal produced by a cosmic particle, was estimated as $\lesssim500$
g/cm$^2$, which corresponds to cosmic rays around 1 PeV. We present a
preliminary discussion of a hypothesis that the TUS161003 event and perhaps
some other bright EAS-like events could be produced by relativistic dust
grains, which were considered a possible component of the cosmic ray flux
beyond the GZK cut-off some time ago.
