Groups acting on dendrons

A dendron is a continuum (a non-empty connected compact Hausdorff space) in which every two distinct points have a separation point. We prove that if a group $G$ acts on a dendron $D$ by homeomorphisms, then either $D$ contains a $G$-invariant subset consisting of one or two points, or $G$ contains a free non-commutative subgroup and, furthermore, the action is strongly proximal.