  The asymptotically flat structure of $\mathcal{N}=(2,0)$ supergravity in
three spacetime dimensions is explored. The asymptotic symmetries are spanned
by an extension of the super-BMS$_3$ algebra, with two independent $\hat{u}(1)$
currents of electric and magnetic type. These currents are associated to $U(1)$
fields being even and odd under parity, respectively. Remarkably, although the
$U(1)$ fields do not generate a backreaction on the metric, they provide
nontrivial Sugawara-like contributions to the BMS$_3$ generators, and hence to
the energy and the angular momentum. The entropy of flat cosmological
spacetimes with $U(1)$ fields then acquires a nontrivial dependence on the
$\hat{u}(1)$ charges. If the spin structure is odd, the ground state
corresponds to Minkowski spacetime, and although the anticommutator of the
canonical supercharges is linear in the energy and in the electric-like
$\hat{u}(1)$ charge, the energy becomes bounded from below by the energy of the
ground state shifted by the square of the electric-like $\hat{u}(1)$ charge. If
the spin structure is even, the same bound for the energy generically holds,
unless the absolute value of the electric-like charge is less than minus the
mass of Minkowski spacetime in vacuum, so that the energy has to be
nonnegative. The explicit form of the Killing spinors is found for a wide class
of configurations that fulfills our boundary conditions, and they exist
precisely when the corresponding bounds are saturated. It is also shown that
the spectra with periodic or antiperiodic boundary conditions for the fermionic
fields are related by spectral flow, in a similar way as it occurs for the
$\mathcal{N}=2$ super-Virasoro algebra. Indeed, our super-BMS$_3$ algebra can
be recovered from the flat limit of the superconformal algebra with
$\mathcal{N}=(2,2)$, truncating the fermionic generators of the right copy.
