  The notion of classical $p$-adic integrable system on a $p$-adic symplectic
manifold was proposed by Voevodsky, Warren and the second author a decade ago
in analogy with the real case. In the present paper we introduce and study,
from the viewpoint of symplectic geometry and topology, the basic properties of
the $p$-adic version of the classical Jaynes-Cummings model. The
Jaynes-Cummings model is a fundamental example of integrable system going back
to the work of Jaynes and Cummings in the 1960s, and which applies to many
physical situations, for instance in quantum optics and quantum information
theory. Several of our results depend on the value of $p$: the structure of the
model depends on the class of the prime $p$ modulo $4$ and $p=2$ requires
special treatment.
