We generalise the dynamic Laplacian introduced in (Froyland, 2015) to a dynamic $p$-Laplacian, in analogy to the generalisation of the standard $2$-Laplacian to the standard $p$-Laplacian for $p>1$. Spectral properties of the dynamic Laplacian are connected to the geometric problem of finding "coherent" sets with persistently small boundaries under dynamical evolution, and we show that the dynamic $p$-Laplacian shares similar geometric connections. In particular, we prove that the first eigenvalue of the dynamic $p$-Laplacian with Dirichlet boundary conditions exists and converges to a dynamic version of the Cheeger constant introduced in (Froyland, 2015) as $p\rightarrow 1$. We develop a numerical scheme to estimate the leading eigenfunctions of the (nonlinear) dynamic $p$-Laplacian, and through a series of examples we investigate the behaviour of the level sets of these eigenfunctions. These level sets define the boundaries of sets in the domain of the dynamics that remain coherent under the dynamical evolution.