  Massive $U(1)$ gauge theories featuring parametrically light vectors are
suspected to belong in the Swampland of consistent EFTs that cannot be embedded
into a theory of quantum gravity. We study four-dimensional, chiral $U(1)$
gauge theories that appear anomalous over a range of energies up to the scale
of anomaly-cancelling massive chiral fermions. We show that such theories
require to be UV-completed at a finite cutoff below which a radial mode must
appear, and cannot be decoupled -- a St\"uckelberg limit does not exist. When
the infrared fermion spectrum contains a mixed $U(1)$-gravitational anomaly,
this class of theories provides a toy model of a boundary into the Swampland,
for sufficiently small values of the vector mass. In this context, we show that
the limit of a parametrically light vector comes at the cost of a quantum
gravity scale that lies parametrically below $M_{Pl}$, and our result provides
field theoretic evidence for the existence of a Swampland of EFTs that is
disconnected from the subset of theories compatible with a gravitational
UV-completion. Moreover, when the low energy theory also contains a $U(1)^3$
anomaly, the Weak Gravity Conjecture scale makes an appearance in the form of a
quantum gravity cutoff for values of the gauge coupling above a certain
critical size.
