  We investigate whether cosmological inflation is possible in a class of flux
compactifications of type IIA strings on rigid Calabi-Yau manifolds, when all
perturbative string corrections are taken into account. We confine ourselves to
the universal hypermultiplet and an abelian vector multiplet, representing
matter in four dimensions. Since all axions can be stabilized by D-instantons,
we choose dilaton and a K\"ahler modulus as the only running scalars. Though
positivity of their scalar potential can be achieved, we find that there is no
slow roll ($\varepsilon > 2$ and $\eta >2$), and no Graceful Exit because the
scalar potential has the run-away behaviour resulting in decompactification. We
conclude that it is impossible to generate phenomenologically viable inflation
in the given class of flux compactifications of type IIA strings, without
explicit breaking of $N=2$ local supersymmetry of the low-energy effective
action.
