  For a fixed Banach operator ideal $\mathcal A$, we use the notion of
$\mathcal A$-compact sets of Carl and Stephani to study $\mathcal A$-compact
polynomials and $\mathcal A$-compact holomorphic mappings. Namely, those
mappings $g\colon X\rightarrow Y$ such that every $x \in X$ has a neighborhood
$V_x$ such that $g(V_x)$ is relatively $\mathcal A$-compact. We show that the
behavior of $\mathcal A$-compact polynomials is determined by its behavior in
any neighborhood of any point. We transfer some known properties of $\mathcal
A$-compact operators to $\mathcal A$-compact polynomials. In order to study
$\mathcal A$-compact holomorphic functions, we appeal to the $\mathcal
A$-compact radius of convergence which allows us to characterize the functions
in this class. Under certain hypothesis on the ideal $\mathcal A$, we give
examples showing that our characterization is sharp.
