  Using a set of general-relativistic magnetohydrodynamics simulations that
include proper neutrino transfer, we assess for the first time the role played
by the fallback accretion onto the remnant from a binary neutron-star merger
over a timescale of hundreds of seconds. In particular, we find that,
independently of the equation of state, the properties of the binary, and the
fate of the remnant, the fallback material reaches a total mass of $\gtrsim
10^{-3}\,M_\odot$, i.e. about $50\%$ of the unbound matter, and that the
fallback accretion rate follows a power-law in time with slope $\sim t^{-5/3}$.
Interestingly, the timescale of the fallback and the corresponding accretion
luminosity are in good agreement with the so-called ``extended emission''
observed in short gamma-ray bursts (GRBs). Using a simple electromagnetic
emission model based on the self-consistent thermodynamical state of the
fallback material heated by r-process nucleosynthesis, we show that this
fallback material can shine in the gamma- and X-rays with luminosities $\gtrsim
\,10^{48}\,{\rm erg/s}$ for hundreds of seconds, thus making it a good and
natural candidate to explain the extended emission in short GRBs. In addition,
our model for the emission by the fallback material reproduces well and rather
naturally some of the phenomenological traits of the extended emission, such as
its softer spectra with respect to the prompt emission and the presence of
exponential cutoffs in time. Our results clearly highlight that fallback flows
onto merger remnants cannot be neglected and the corresponding emission
represents a very promising and largely unexplored avenue to explain the
complex phenomenology of GRBs.
