  We present possible realizations of gauge inflation arising from a 5D ${\cal
N}=1$ supersymmetric U(1) model, where the extra dimension is compactified on a
circle. A one-loop inflaton effective 4D potential is generated, with the
inflaton being a 'Wilson-line field'. It relies on a SUSY breaking. We first
consider SUSY breaking to occur spontaneously within a 'no-scale' model by a
non zero $F$-term of the radion superfield which transmits SUSY breaking into
the 'visible' sector. As an alternative, we study $D$-term SUSY breaking
originating directly from the 5D gauge supermultiplet. Together with the usual
KK resummation method, we present a calculation with the world-line formalism.
The latter allows one to get the resulting effective potential directly as a
sum over all winding modes. For both presented scenarios, the generated
effective potentials have suitable forms for realizing successful inflation,
i.e. are flat enough and give the needed number of ${\rm e}$-foldings. In
addition, there is a natural way to get strongly suppressed values for the
potentials, which then could be associated with dark energy/quintessence.
