  We construct the first analytic examples of topologically non-trivial
solutions of the (3+1)-dimensional $U(1)$ gauged Skyrme model within a finite
box in (3+1)-dimensional flat space-time. There are two types of gauged
solitons. The first type corresponds to gauged Skyrmions living within a finite
volume. The second corresponds to gauged time-crystals (smooth solutions of the
$U(1)$ gauged Skyrme model whose periodic time-dependence is protected by a
winding number). The notion of electromagnetic duality can be extended for
these two types of configurations in the sense that the electric and one of the
magnetic components can be interchanged. These analytic solutions show very
explicitly the Callan-Witten mechanism (according to which magnetic monopoles
may "swallow" part of the topological charge of the Skyrmion) since the
electromagnetic field contribute directly to the conserved topological charge
of the gauged Skyrmions. As it happens in superconductors, the magnetic field
is suppressed in the core of the gauged Skyrmions. On the other hand, the
electric field is strongly suppresed in the core of gauged time crystals.
