  We construct approximate inflationary solutions rolling away from the
unstable maximum of p-adic string theory, a nonlocal theory with derivatives of
all orders. Novel features include the existence of slow-roll solutions even
when the slow-roll parameters, as usually defined, are much greater than unity,
as well as the need for the Hubble parameter to exceed the string mass scale
m_s. We show that the theory can be compatible with CMB observations if g_s /
\sqrt{p} ~ 10^{-7}, where g_s is the string coupling, and if m_s < 10^{-6} M_p.
A red-tilted spectrum is predicted, and the scalar-to-tensor ratio is bounded
from above as r < 0.006. The p-adic theory is shown to have identical
inflationary predictions to a local theory with superPlanckian parameter
values, but with the advantage that the p-adic theory is ultraviolet complete.
