  We construct a functorial decomposition of de Rham cohomology sheaves, called
weight decomposition, for smooth analytic spaces over non-Archimedean fields
embeddable into $\mathbf{C}_p$, which generalizes a construction of Berkovich
and solves a question raised by him. We then investigate complexes of real
tropical differential forms and currents introduced by Chambert-Loir and
Ducros, by establishing a relation with the weight decomposition and defining
tropical cycle maps with values in the corresponding Dolbeault cohomology. As
an application, we show that algebraic cycles that are cohomologically trivial
in the algebraic de Rham cohomology are cohomologically trivial in the
Dolbeault cohomology of currents as well.
