  Let $E$ and $F$ be complex Banach spaces, $U$ be an open subset of $E$ and
$1\leq p\leq\infty$. We introduce and study the notion of a Cohen strongly
$p$-summing holomorphic mapping from $U$ to $F$, a holomorphic version of a
strongly $p$-summing linear operator. For such mappings, we establish both
Pietsch domination/factorization theorems and analyse their linearizations from
$\mathcal{G}^\infty(U)$ (the canonical predual of $\mathcal{H}^\infty(U)$) and
their transpositions on $\mathcal{H}^\infty(U)$. Concerning the space
$\mathcal{D}_p^{\mathcal{H}^\infty}(U,F)$ formed by such mappings and endowed
with a natural norm $d_p^{\mathcal{H}^\infty}$, we show that it is a regular
Banach ideal of bounded holomorphic mappings generated by composition with the
ideal of strongly $p$-summing linear operators. Moreover, we identify the space
$(\mathcal{D}_p^{\mathcal{H}^\infty}(U,F^*),d_p^{\mathcal{H}^\infty})$ with the
dual of the completion of tensor product space $\mathcal{G}^\infty(U)\otimes F$
endowed with the Chevet--Saphar norm $g_p$.
