  Chambert-Loir and Ducros have recently introduced a theory of real valued
differential forms and currents on Berkovich spaces. In analogy to the theory
of forms with logarithmic singularities, we enlarge the space of differential
forms by so called delta-forms on the non-archimedean analytification of an
algebraic variety. This extension is based on an intersection theory for
tropical cycles with smooth weights. We prove a generalization of the
Poincar\'e-Lelong formula which allows us to represent the first Chern current
of a formally metrized line bundle by a delta-form. We introduce the associated
Monge-Amp\`ere measure $\mu$ as a wedge-power of this first Chern delta-form
and we show that $\mu$ is equal to the corresponding Chambert-Loir measure. The
star-product of Green currents is a crucial ingredient in the construction of
the arithmetic intersection product. Using the formalism of delta-forms, we
obtain a non-archimedean analogue at least in the case of divisors. We use it
to compute non-archimedean local heights of proper varieties.
