DIAMONDS: a new Bayesian Nested Sampling tool. Application to Peak Bagging of solar-like oscillations. To exploit the full potential of Kepler light curves, sophisticated and robust analysis tools are now required more than ever. Characterizing single stars with an unprecedented level of accuracy and subsequently analyzing stellar populations in detail are fundamental to further constrain stellar structure and evolutionary models. We developed a new code, termed Diamonds, for Bayesian parameter estimation and model comparison by means of the nested sampling Monte Carlo (NSMC) algorithm, an efficient and powerful method very suitable for high-dimensional and multi-modal problems. A detailed description of the features implemented in the code is given with a focus on the novelties and differences with respect to other existing methods based on NSMC. Diamonds is then tested on the bright F8 V star KIC 9139163, a challenging target for peak-bagging analysis due to its large number of oscillation peaks observed, which are coupled to the blending that occurs between ℓ=2,0 peaks, and the strong stellar background signal. We further strain the performance of the approach by adopting a 1147.5 days-long Kepler light curve. The Diamonds code is able to provide robust results for the peak-bagging analysis of KIC 9139163. We test the detection of different astrophysical backgrounds in the star and provide a criterion based on the Bayesian evidence for assessing the peak significance of the detected oscillations in detail. We present results for 59 individual oscillation frequencies, amplitudes and linewidths and provide a detailed comparison to the existing values in the literature. Lastly, we successfully demonstrate an innovative approach to peak bagging that exploits the capability of Diamonds to sample multi-modal distributions, which is of great potential for possible future automatization of the analysis technique.