  We initiate the study of fine $p$-(super)minimizers, associated with
$p$-harmonic functions, on finely open sets in metric spaces, where $1 < p <
\infty$. After having developed their basic theory, we obtain the $p$-fine
continuity of the solution of the Dirichlet problem on a finely open set with
continuous Sobolev boundary values, as a by-product of similar pointwise
results. These results are new also on unweighted $\mathbf{R}^n$. We build this
theory in a complete metric space equipped with a doubling measure supporting a
$p$-Poincar\'e inequality.
