  We provide a sharp $C^{1,\alpha}$ estimate up to the boundary for a viscosity
solution of a degenerate fully nonlinear elliptic equation with the oblique
boundary condition on a $C^1$ domain. To this end, we first obtain a uniform
boundary H{\"o}lder estimate with the oblique boundary condition in an "almost
$C^1$-flat" domain for the equations which is uniformly elliptic only where the
gradient is far from some point, and then we establish a desired $C^{1,\alpha}$
regularity based on perturbation and compactness arguments.
