A flat Friedmann-Robertson-Walker universe dominated by a cosmological constant (Λ) and cold dark matter (CDM) has been the working model preferred by cosmologists since the discovery of cosmic acceleration. However, tensions of various degrees of significance are known to be present among existing datasets within the ΛCDM framework. In particular, the Lyman-α forest measurement of the baryon acoustic oscillations (BAO) by the Baryon Oscillation Spectroscopic Survey3 prefers a smaller value of the matter density fraction ΩM than that preferred by cosmic microwave background (CMB). Also, the recently measured value of the Hubble constant, H0=73.24±1.74 km s-1 Mpc-1 (ref.), is 3.4σ higher than the 66.93 ± 0.62 km s-1 Mpc-1 inferred from the Planck CMB data7. In this work, we investigate whether these tensions can be interpreted as evidence for a non-constant dynamical dark energy. Using the Kullback-Leibler divergence to quantify the tension between datasets, we find that the tensions are relieved by an evolving dark energy, with the dynamical dark energy model preferred at a 3.5σ significance level based on the improvement in the fit alone. While, at present, the Bayesian evidence for the dynamical dark energy is insufficient to favour it over ΛCDM, we show that, if the current best-fit dark energy happened to be the true model, it would be decisively detected by the upcoming Dark Energy Spectroscopic Instrument survey.