  We study the regularity of solutions to the fully nonlinear thin obstacle
problem. We establish local $C^{1,\alpha}$ estimates on each side of the smooth
obstacle, for some small $\alpha > 0$.
  Our results extend those of Milakis-Silvestre in two ways: first, we do not
assume solutions nor operators to be symmetric, and second, our estimates are
local, in the sense that do not rely on the boundary data.
  As a consequence, we prove $C^{1,\alpha}$ regularity even when the problem is
posed in general Lipschitz domains.
