  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.
