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In this note we are concerned with interior regularity properties of the$p$-Poisson problem $\Delta_p(u)=f$ with $p>2$. For all $0<\lambda\leq 1$ weconstuct right-hand sides $f$ of differentiability $-1+\lambda$ such that the(Besov-) smoothness of corresponding solutions $u$ is essentially limited to$1+\lambda / (p-1)$. The statements are of local nature and cover allintegrability parameters. They particularly imply the optimality of a shifttheorem due to Savar\'e [J. Funct. Anal. 152:176-201, 1998], as well as of somerecent Besov regularity results of Dahlke et al. [Nonlinear Anal. 130:298-329,2016].  Keywords: Nonlinear and adaptive approximation, Besov space, regularity ofsolutions, $p$-Poisson problem.

On the lack of interior regularity of the $p$-Poisson problem with $p>2$