The 70-month Swift-BAT catalogue provides a sensitive view of the extragalactic X-ray sky at hard energies (>10 keV) containing about 800 active galactic nuclei (AGN). We explore its content in heavily obscured, Compton-thick AGN by combining the BAT (14−195 keV) with the lower energy XRT (0.3−10 keV) data. We apply a Bayesian methodology using Markov chains to estimate the exact probability distribution of the column density for each source. We find 53 possible Compton-thick sources (probability range 3−100%) translating to a ~7% fraction of the AGN in our sample. We derive the first parametric luminosity function of Compton-thick AGN. The unabsorbed luminosity function can be represented by a double power law with a break at L⋆ ~ 2 × 1042erg s-1 in the 20−40 keV band. The Compton-thick AGN contribute ~17% of the total AGN emissivity. We derive an accurate Compton-thick number count distribution taking into account the exact probability of a source being Compton-thick and the flux uncertainties. This number count distribution is critical for the calibration of the X-ray background synthesis models, i.e. for constraining the intrinsic fraction of Compton-thick AGN. We find that the number counts distribution in the 14−195 keV band agrees well with our models which adopt a low intrinsic fraction of Compton-thick AGN (~ 12%) among the total AGN population and a reflected emission of ~ 5%. In the extreme case of zero reflection, the number counts can be modelled with a fraction of at most 30% Compton-thick AGN of the total AGN population and no reflection. Moreover, we compare our X-ray background synthesis models with the number counts in the softer 2−10 keV band. This band is more sensitive to the reflected component and thus helps us to break the degeneracy between the fraction of Compton-thick AGN and the reflection emission. The number counts in the 2−10 keV band are well above the models which assume a 30% Compton-thick AGN fraction and zero reflection, while they are in better agreement with models assuming 12% Compton-thick fraction and 5% reflection. The only viable alternative for models invoking a high number of Compton-thick AGN is to assume evolution in their number with redshift. For example, in the zero reflection model the intrinsic fraction of Compton-thick AGN should rise from 30% at redshift z ~ 0 to about 50% at a redshift of z = 1.1.