We investigate stationary rotationally symmetric solutions of a particular truncation of Horndeski theory in three dimensions, including a non-minimal scalar kinetic coupling to the curvature. After discussing the special case of a vanishing scalar charge, which includes most of the previously known solutions, we reduce the general case to an effective mechanical model in a three-dimensional target space. We analyze the possible near-horizon behaviors, and conclude that black hole solutions with degenerate horizons and constant curvature asymptotics may exist if the minimal and non-minimal scalar coupling constants have the same sign. In a special case, we find a new analytic rotating black hole solution with scalar hair and degenerate horizon. This is geodesically and causally complete, and asymptotic to the extreme BTZ metric. We also briefly discuss soliton solutions in another special case.