  We consider the normalized $p$-Poisson problem $$-\Delta^N_p u=f \qquad
\text{in}\quad \Omega.$$ The normalized $p$-Laplacian $\Delta_p^{N}u:=|D
u|^{2-p}\Delta_p u$ is in non-divergence form and arises for example from
stochastic games. We prove $C^{1,\alpha}_{loc}$ regularity with nearly optimal
$\alpha$ for viscosity solutions of this problem. In the case $f\in
L^{\infty}\cap C$ and $p>1$ we use methods both from viscosity and weak theory,
whereas in the case $f\in L^q\cap C$, $q>\max(n,\frac p2,2)$, and $p>2$ we rely
on the tools of nonlinear potential theory.
