  We obtain a sharp estimate on the norm of the differential of a harmonic map
from the unit disc $\mathbb D$ in $\mathbb C$ into the unit ball $\mathbb B^n$
in $\mathbb R^n$, $n\ge 2$, at any point where the map is conformal. In
dimension $n=2$, this generalizes the classical Schwarz-Pick lemma, and for
$n\ge 3$ it gives the optimal Schwarz-Pick lemma for conformal minimal discs
$\mathbb D\to \mathbb B^n$. This implies that conformal harmonic immersions $M
\to \mathbb B^n$ from any hyperbolic conformal surface are distance-decreasing
in the Poincar$\mathrm{\'e}$ metric on $M$ and the Cayley-Klein metric on the
ball $\mathbb B^n$, and the extremal maps are precisely the conformal
embeddings of the disc $\mathbb D$ onto affine discs in $\mathbb B^n$. By using
these results, we lay the foundations of the hyperbolicity theory for domains
in $\mathbb R^n$ based on minimal surfaces.
