  In this work, we study the supersingular locus of the Shimura variety
associated to the unitary group $\mathrm{GU}(2,4)$ over a ramified prime. We
show that the associated Rapoport-Zink space is flat, and we give an explicit
description of the irreducible components of the reduction modulo $p$ of the
basic locus. In particular, we show that these are universally homeomorphic to
either a generalized Deligne-Lusztig variety for a symplectic group or to the
closure of a vector bundle over a classical Deligne-Lusztig variety for an
orthogonal group. Our results are confirmed in the group-theoretical setting by
the reduction method \`a la Deligne and Lusztig and the study of the admissible
set.
