  We give the p-adic and F_q((t)) analogue of the real van der Corput Lemma,
where the real condition of sufficient smoothness for the phase is replaced by
the condition that the phase is a convergent power series. This van der Corput
style result allows us, in analogy to the real situation, to study singular
Fourier transforms on suitably curved (analytic) manifolds and opens the way
for further applications. As one such further application we give the
restriction theorem for Fourier transforms of L^p functions to suitably curved
analytic manifolds over non-archimedean local fields, similar to the real
restriction result by E. Stein and C. Fefferman.
