  Asymptotic symmetries of electric and magnetic Carrollian gravitational
theories with a negative cosmological constant $\Lambda$ are analyzed in 3+1
space-time dimensions. In the magnetic theory, the asymptotic symmetry algebra
is given by the conformal Carroll algebra in three dimensions, which is
infinite-dimensional and isomorphic to the BMS$_{4}$ algebra. These results are
in full agreement with holographic expectations, providing a new framework for
the study of "Carrollian holography." On the contrary, in the case of the
electric theory, the presence of a negative $\Lambda$ turns out to be
incompatible with a consistent set of asymptotic conditions, that can be traced
back to the absence of a sensible ground state configuration. This can be
improved if the Carrollian theory obtained from an electric contraction of
Euclidean General Relativity is considered. In this case, asymptotic conditions
can be constructed with an asymptotic symmetry algebra given by
$so\left(1,4\right)$. However, it is shown that the space of spherically
symmetric solutions of this theory is degenerate.
