The discovery potential from astronomical and other data is limited by their noise. We introduce a novel non-parametric noise reduction technique based on Bayesian inference techniques, fully adaptive Bayesian algorithm for data analysis (FABADA) that automatically improves the signal-to-noise ratio of one- and two-dimensional data, such as astronomical images and spectra. The algorithm iteratively evaluates possible smoothed versions of the data, the smooth models, estimating the underlying signal that is statistically compatible with the noisy measurements. Iterations stop based on the evidence and the χ2 statistic of the last smooth model. We then compute the expected value of the signal as a weighted average of the whole set of smooth models. We explain the mathematical formalism and numerical implementation of the algorithm, and evaluate its performance in terms of the peak signal-to-noise ratio, the structural similarity index, and the time payload, using a battery of real astronomical observations. Our FABADA yields results that, without any parameter tuning, are comparable with standard image processing algorithms whose parameters have been optimized based on the true signal to be recovered, something that is impossible in a real application. On the other hand, state-of-the-art non-parametric methods, such as block-matching and three-dimensional filtering, offer slightly better performance at high signal-to-noise ratio, while our algorithm is significantly more accurate for extremely noisy data, a situation usually encountered in astronomy.