  The baby Skyrme model is a well-known nonlinear field theory supporting
topological solitons in two space dimensions. In the limit where the term
quadratic in derivatives (the "sigma model term") vanishes some additional
structure emerges. The resulting ("extreme" or "restricted" or "BPS") baby
Skyrme model has exact soliton solutions saturating a BPS bound which exists
for this restricted model. Further, the restricted model has infinitely many
symmetries and infinitely many conservation laws. Here we consider the gauged
version of the restricted baby Skyrme model with gauge group U(1) and the usual
Maxwell term for the gauge field. We find that, again, there exists a BPS bound
and BPS solutions saturating this bound. We further find that the whole problem
is essentially determined by a new kind of superpotential equation. The BPS
bound and the corresponding BPS solitons only may exist for potentials such
that the superpotential equation has a solution which exists globally, i.e., on
the whole target space. We also calculate soliton solutions both exactly and
numerically, completely confirming our qualitative analytical results.
