  This paper is concerned with the Dirichlet eigenvalue problem associated to
the $\infty$-Laplacian in metric spaces. We establish a direct PDE approach to
find the principal eigenvalue and eigenfunctions in a proper geodesic space
without assuming any measure structure. We provide an appropriate notion of
solutions to the $\infty$-eigenvalue problem and show the existence of
solutions by adapting Perron's method. Our method is different from the
standard limit process via the variational eigenvalue formulation for
$p$-Laplacian in the Euclidean space.
