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Let L be an ample line bundle on a smooth projective variety X over anon-archimedean field K. For a continuous metric on L, we show in the followingtwo cases that the semipositive envelope is a continuous semipositive metric onL 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 theassumption that X is a surface defined geometrically over the function field ofa curve over a perfect field k of positive characteristic. The second caseholds in higher dimensions if we assume resolution of singularities over k. Theproof follows a strategy from Boucksom, Favre and Jonsson, replacing multiplierideals by test ideals. Finally, the appendix by Burgos and Sombra provides anexample of a semipositive metric whose retraction is not semipositive. Theexample is based on the construction of a toric variety which has twoSNC-models which induce the same skeleton but different retraction maps.

Continuity of Plurisubharmonic Envelopes in Non-Archimedean Geometry and  Test Ideals (with an Appendix by Jos\'e Ignacio Burgos Gil and Mart\'in  Sombra)