  We initiate the study of p-adic algebraic groups G from the
stability-theoretic and definable topological-dynamical points of view, that
is, we consider invariants of the action of G on its space of types over Q_p in
the language of fields. We consider the additive and multiplicative groups of
Q_p and Z_p, the group of upper triangular invertible 2\times 2 matrices,
SL(2,Z_p), and, our main focus, SL(2,Q_p). In all cases we identify f-generic
types (when they exist), minimal subflows, and idempotents. Among the main
results is that the ``Ellis group" of SL(2,Q_p)$ is the profinite completion of
Z, yielding a counterexample to Newelski's conjecture with new features: G =
G^{00} = G^{000} but the Ellis group is infinite. A final section deals with
the action of SL(2,Q_p) on the type-space of the projective line over Q_p.
