  Let L be an ample line bundle on a smooth projective variety X over a
non-archimedean field K. For a continuous metric on L, we show in the following
two cases that the semipositive envelope is a continuous semipositive metric on
L and that the non-archimedean Monge-Amp\`ere equation has a solution. First,
we prove it for curves using results of Thuillier. Second, we show it under the
assumption that X is a surface defined geometrically over the function field of
a curve over a perfect field k of positive characteristic. The second case
holds in higher dimensions if we assume resolution of singularities over k. The
proof follows a strategy from Boucksom, Favre and Jonsson, replacing multiplier
ideals by test ideals. Finally, the appendix by Burgos and Sombra provides an
example of a semipositive metric whose retraction is not semipositive. The
example is based on the construction of a toric variety which has two
SNC-models which induce the same skeleton but different retraction maps.
