  This paper gives foundations of a theory of $p$-adic cyclotomic multiple zeta
values by explicit formulas, and a new notion called pro-unipotent harmonic
actions. We prove formulas which express $p$-adic cyclotomic multiple zeta
values in terms of certain cyclotomic multiple harmonic sums and vice-versa ;
these formulas keep track of the motivic Galois action via the pro-unipotent
harmonic actions. The proof uses the main result of \cite{I-1}. One of the
formulas proves a conjecture of Akagi, Hirose and Yasuda which sheds light on
the finite multiple zeta values of Kaneko and Zagier.
