  Given a bounded finely open set $V$ and a function $f$ on the fine boundary
of $V$, we introduce four types of upper Perron solutions to the nonlinear
Dirichlet problem for $p$-energy minimizers, $1<p<\infty$, with $f$ as boundary
data. These solutions are given as pointwise infima of suitable families of
fine $p$-superminimizers in $V$. We show (under natural assumptions) that the
four upper Perron solutions are equal quasieverywhere and that they are fine
$p$-minimizers of the $p$-energy integral. We moreover show that the upper and
lower Perron solutions coincide quasieverywhere for Sobolev and for uniformly
continuous boundary data, i.e.\ that such boundary data are resolutive. For the
uniformly continuous boundary data, the Perron solutions are also shown to be
finely continuous and thus finely $p$-harmonic. We prove our results in a
complete metric space $X$ equipped with a doubling measure supporting a
$p$-Poincar\'e inequality, but they are new also in unweighted $\mathbf{R}^n$.
