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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 holomorphicmappings on an open subset $U$ of a complex Banach space $E$ whose ranges arerelatively $p$-compact subsets of $F$. We characterize such holomorphicmappings as those whose Mujica's linearisations on the canonical predual of$\mathcal{H}^\infty(U)$ are $p$-compact operators. This fact allows us to makea complete study of them. We show that $\mathcal{H}^\infty_{\mathcal{K}_p}$ isa surjective Banach ideal of bounded holomorphic mappings which is generated bycomposition with the ideal of $p$-compact operators and contains the Banachideal of all right $p$-nuclear holomorphic mappings. We also characterizeholomorphic mappings with relatively $p$-compact ranges as those boundedholomorphic 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 $\ell_p$).

On holomorphic mappings with relatively $p$-compact range