  The purpose of this paper is to prove integrality for certain $p$-adic
iterated Coleman integrals. As underlying geometry we will take the complement
of a divisor $D\subset X$ with good reduction, where $X$ is the projective line
or an elliptic curve over the Witt vectors of a perfect characteristic $p$
field. As a corollary we prove a lower bound for the valuations of $p$-adic
multiple zeta values.
