Various methods have recently been proposed to estimate causal effects with confidence intervals that are uniformly valid over a set of data-generating processes when high-dimensional nuisance models are estimated by post-model-selection or machine learning estimators. These methods typically require that all the confounders are observed to ensure identification of the effects. We contribute by showing how valid semiparametric inference can be obtained in the presence of unobserved confounders and high-dimensional nuisance models. We propose uncertainty intervals that allow for unobserved confounding, and show that the resulting inference is valid when the amount of unobserved confounding is not arbitrarily large; the latter is formalized in terms of convergence rates. Simulation experiments illustrate the finite sample properties of the proposed intervals. Finally, a case study on the effect of smoking during pregnancy on birth weight is used to illustrate the use of the methods introduced to perform an informed sensitivity analysis to the presence of unobserved confounding.