  Let $M$ be a $C^2$-smooth Riemannian manifold with boundary and $N$ a
complete $C^2$-smooth Riemannian manifold. We show that each stationary
$p$-harmonic mapping $u\colon M\to N$, whose image lies in a compact subset of
$N$, is locally $C^{1,\alpha}$ for some $\alpha\in (0,1)$, provided that $N$ is
simply connected and has non-positive sectional curvature. We also prove
similar results for each minimizing $p$-harmonic mapping $u\colon M\to N$ with
$u(M)$ being contained in a regular geodesic ball. Moreover, when $M$ has
non-negative Ricci curvature and $N$ is simply connected and has non-positive
sectional curvature, we deduce a quantitative gradient estimate for each
$C^1$-smooth weakly $p$-harmonic mapping $u\colon M\to N$. Consequently, we
obtain a Liouville-type theorem for $C^1$-smooth weakly $p$-harmonic mappings
in the same setting.
