On integrality of $p$-adic iterated integrals

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.