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.