We show the existence of chaotic (in the sense of Devaney) polynomials on Banach spaces of $q$-summable sequences. Such polynomials $P$ consist of composition of the backward shift with a certain fixed polynomial $p$ of one complex variable on each coordinate. In general we also prove that $P$ is chaotic in the sense of Auslander and Yorke if and only if $0$ belongs to the Julia set of $p$.