  We study p-adic hyper-Kloosterman sums, a generalization of the Kloosterman
sum with a parameter k that recovers the classical Kloosterman sum when k=2,
over general p-adic rings and even equal characteristic local rings. These can
be evaluated by a simple stationary phase estimate when k is not divisible by
p, giving an essentially sharp bound for their size. We give a more complicated
stationary phase estimate to evaluate them in the case when k is divisible by
p. This gives both an upper bound and a lower bound showing the upper bound is
essentially sharp. This generalizes previously known bounds of Cochrane, Liu,
and Zhen in the case of $\mathbb Z_p$. The lower bounds in the equal
characteristic case have two applications to function field number theory,
showing that certain short interval sums and certain moments of Dirichlet
L-functions do not, as one might hope, admit square-root cancellation.
