The search for new exoplanets by direct imaging is a very active research topic in astronomy. The detection is particularly challenging because of the very high contrast between the host star and the companions. They thus remain hidden by a nonstationary background displaying strong spatial correlations. We propose a new algorithm named PACO (for PAtch COvariances) for reduction of differential imaging datasets. Contrary to existing approaches, we model the background correlations using a local Gaussian distribution that locally captures the spatial correlations at the scale of a patch of a few tens of pixels. The decision in favor of the presence or the absence of an exoplanet in then performed by a binary hypothesis test. The method is completely parameter-free and produces both stationary and statistically grounded detection maps so that the false alarm rate, the probability of detection and the contrast can be directly assessed without post-processing and/or Monte-Carlo simulations. We describe in a forthcoming paper its detailed principle and implementation. In this paper, we recall the principle of the PACO algorithm and we give new illustrations of its benefits in terms of detection capabilities on datasets from the VLT/SPHERE-IRDIS instrument. We also apply our algorithm on multi-spectral datasets from the VLT/SPHERE-IFS spectrograph. The performance of PACO is compared to state-of-the-art algorithms such as TLOCI and KLIP-PCA.