We present a data-driven approach for probabilistic wind power forecasting based on conditional normalizing flow~(CNF). In contrast with the existing, this approach is distribution-free (as for non-parametric and quantile-based approaches) and can directly yield continuous probability densities, hence avoiding quantile crossing. It relies on a base distribution and a set of bijective mappings. Both the shape parameters of the base distribution and the bijective mappings are approximated with neural networks. Spline-based conditional normalizing flow is considered owing to its universal approximation capability. Over the training phase, the model sequentially maps input examples onto samples of base distribution, where parameters are estimated through maximum likelihood. To issue probabilistic forecasts, one eventually map samples of the base distribution into samples of a desired distribution. Case studies based on open datasets validate the effectiveness of the proposed model, and allows us to discuss its advantages and caveats with respect to the state of the art. Code will be released upon publication.