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We consider the normalized $p$-Poisson problem $$-\Delta^N_p u=f \qquad\text{in}\quad \Omega.$$ The normalized $p$-Laplacian $\Delta_p^{N}u:=|Du|^{2-p}\Delta_p u$ is in non-divergence form and arises for example fromstochastic games. We prove $C^{1,\alpha}_{loc}$ regularity with nearly optimal$\alpha$ for viscosity solutions of this problem. In the case $f\inL^{\infty}\cap C$ 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 relyon the tools of nonlinear potential theory.

$C^{1,\alpha}$ regularity for the normalized $p$-Poisson problem