  The Perron method for solving the Dirichlet problem for $p$-harmonic
functions is extended to unbounded open sets in the setting of a complete
metric space with a doubling measure supporting a $p$-Poincar\'e inequality,
$1<p<\infty$. The upper and lower ($p$-harmonic) Perron solutions are studied
for $p$-parabolic open sets. It is shown that continuous functions and
quasicontinuous Dirichlet functions are resolutive (i.e., that their upper and
lower Perron solutions coincide) and that the Perron solution coincides with
the $p$-harmonic extension. It is also shown that Perron solutions are
invariant under perturbation of the function on a set of capacity zero.
