Let $X$ and $Y$ be Banach spaces and let $1\leq p<\infty$. An $m$-homogeneous polynomial $P:X\to Y$ is $p$-limited if $P(B_X)$ is a $p$-limited subset of $Y$, and coarse $p$-limited if $P(B_X)$ is a coarse $p$-limited subset of $Y$. We prove that the projective tensor product of $p$-limited operators is $p$-limited, whereas the analogous result does not hold for coarse $p$-limited operators. We also show that the Aron-Berner extension of a $p$-limited polynomial remains $p$-limited. Furthermore, we define and study $p$-limited/coarse $p$-limited holomorphic mappings and present examples to illustrate the behavior of such mappings.