  Related to the concept of $p$-compact operator with $p\in [1,\infty]$
introduced by Sinha and Karn, this paper deals with the space
$\mathcal{H}^\infty_{\mathcal{K}_p}(U,F)$ of all Banach-valued holomorphic
mappings on an open subset $U$ of a complex Banach space $E$ whose ranges are
relatively $p$-compact subsets of $F$. We characterize such holomorphic
mappings as those whose Mujica's linearisations on the canonical predual of
$\mathcal{H}^\infty(U)$ are $p$-compact operators. This fact allows us to make
a complete study of them. We show that $\mathcal{H}^\infty_{\mathcal{K}_p}$ is
a surjective Banach ideal of bounded holomorphic mappings which is generated by
composition with the ideal of $p$-compact operators and contains the Banach
ideal of all right $p$-nuclear holomorphic mappings. We also characterize
holomorphic mappings with relatively $p$-compact ranges as those bounded
holomorphic mappings which factorize through a quotient space of $\ell_{p^*}$
or as those whose transposes are quasi $p$-nuclear operators (respectively,
factor through a closed subspace of $\ell_p$).
