Group actions on $p$-adic symplectic manifolds

Let $p$ be a prime number. We introduce symplectic actions of $p$-adic analytic Lie groups on $p$-adic symplectic manifolds. Then we show that any $p$-adic symplectic action $G\times (M,\omega )\to (M,\omega )$ has a momentum map $\mu :M\to {\mathfrak{g}}^{\ast }$, and that a proper $p$-adic symplectic action is Hamiltonian if and only if every orbit is isotropic. We conclude by defining $p$-adic symplectic toric manifolds, by analogy with the real case.