The inclusive double-spin asymmetry, A1p, has been measured at COMPASS in deep-inelastic polarised muon scattering off a large polarised NH3 target. The data, collected in the year 2007, cover the range Q2>1(GeV/c)2, 0.004<x<0.7 and improve the statistical precision of g1p(x) by a factor of two in the region x<0.02. The new proton asymmetries are combined with those previously published for the deuteron to extract the non-singlet spin-dependent structure function g1NS(x,Q2). The isovector quark density, Δq3(x,Q2), is evaluated from a NLO QCD fit of g1NS. The first moment of Δq3 is in good agreement with the value predicted by the Bjorken sum rule and corresponds to a ratio of the axial and vector coupling constants |gA/gV|=1.28±0.07(stat.)±0.10(syst.).