We constrain the basic cosmological parameters using the first observations by the Very Small Array (VSA) in its extended configuration, together with existing cosmic microwave background data and other cosmological observations. We estimate cosmological parameters for four different models of increasing complexity. In each case, careful consideration is given to implied priors and the Bayesian evidence is calculated in order to perform model selection. We find that the data are most convincingly explained by a simple flat ΛCDM cosmology without tensor modes. In this case, combining just the VSA and COBE data sets yields the 68 per cent confidence intervals Ωbh2= 0.034, Ωdmh2= 0.18, h= 0.72, ns= 1.07 and σ8= 1.17. The most general model considered includes spatial curvature, tensor modes, massive neutrinos and a parametrized equation of state for the dark energy. In this case, by combining all recent cosmological data, we find, in particular, a 95 per cent limit on the tensor-to-scalar ratio R < 0.63 and on the fraction of massive neutrinos fν < 0.11; we also obtain the 68 per cent confidence interval w=−1.06 on the equation of state of dark energy.