The Laplace-Beltrami operators on the unit ball and the half-space \(\{x_ p>0\}\) of \(R^ p\) are defined respectively by \[ \Delta_ 0={1- \| x\|^ 2\over 4}\cdot\left(\Delta+{2(p-2)\over 1-\| x\|^ 2}\cdot\langle x,\text{grad}\rangle\right)\text{ and } \Delta_ +=x^ 2_ p\cdot\left(\Delta+{2-p\over x_ p}\cdot{\partial\over\partial x_ p}\right), \] and solutions are called hyperbolic-harmonic functions. They share several properties of harmonic functions, including the existence of a Poisson formula, and they have the advantage that their class is stable under composition on the right with Möbius maps. For \(p=2\), they are just the usual harmonic functions. In this paper, the author obtains a version of the Schwarz lemma for harmonic and hyperbolic-harmonic functions on the unit ball: if \(| h|<1\) and \(h(0)=a\), he obtains sharp upper and lower bounds for \(| h(x)|\) in terms of \(\| x\|\) and \(a\). These are expressed as certain spherical integrals of the Poisson kernel and can be evaluated. Similar bounds are obtained for \(\|(\text{grad} h)(0)\|\). For the hyperbolic-harmonic case this can be transferred to the half-space using Möbius maps.