A version of the fundamental Calabi-Yau theorem in complex analytic geometry states that if $L$ is an ample line bundle on a smooth, complex projective variety $X$, then any smooth volume form on $X$ of the same total mass as $L$ is the curvature volume form of a unique smooth metric on $L$. Recently Yuan and Zhang proved a version of the uniqueness statement in non-Archimedean geometry. I will report on joint work with S. Boucksom and C. Favre where we prove the corresponding existence result. Our method requires us to use a notion of singular semipositive metrics in a non-Archimedean context.