  Diffeomorphism groups $G$ of manifolds $M$ on locally $\bf F$-convex spaces
over non-Archimedean fields $\bf F$ are investigated. It is shown that their
structure has many differences with the diffeomorphism groups of real and
complex manifolds. It is proved that $G$ is not a Banach-Lie group, but it has
a neighbourhood $W$ of the unit element $e$ such that each element $g$ in $W$
belongs to at least one corresponding one-parameter subgroup.
  It is proved that $G$ is simple and perfect. Its compact subgroups $G_c$ are
studied such that a dimension over $\bf F$ of its tangent space $dim_{\bf
F}T_eG_c$ in $e$ may be infinite. This is used for decompositions of continuous
representations into irreducible and investigations of induced representations.
