  We study the dynamics of the H\'enon map defined over complete, locally
compact non-Archimedean fields of odd residue characteristic. We establish
basic properties of its one-sided and two-sided filled Julia sets, and we
determine, for each H\'enon map, whether these sets are empty or nonempty,
whether they are bounded or unbounded, and whether they are equal to the unit
ball or not. On a certain region of the parameter space we show that the filled
Julia set is an attractor. We prove that, for infinitely many distinct H\'enon
maps over ${\mathbb Q}_3$, this attractor is infinite and supports an SRB-type
measure describing the distribution of all nearby forward orbits. We include
some numerical calculations which suggest the existence of such infinite
attractors over ${\mathbb Q}_5$ and ${\mathbb Q}_7$ as well. On a different
region of the parameter space, we show that the H\'enon map is topologically
conjugate on its filled Julia set to the two-sided shift map on the space of
bisequences in two symbols.
