  We generalize the definition of overconvergent $p$-adic multiple
polylogarithms and of $p$-adic cyclotomic multiple zeta values and we prove a
bound on their norm. A byproduct of the proof is a characterization of these
objects in terms of certain regularized $p$-adic iterated integrals. The
generalization of the definition consists in replacing the underlying Frobenius
structure by its iterations. The bound on the norms of overconvergent $p$-adic
multiple polylogarithms that we obtain is a prerequisite for our subsequent
papers on $p$-adic cyclotomic multiple zeta values.
