We present a precise measurement of the proton longitudinal double-spin asymmetry A(1)(p) and the proton spin-dependent structure function g(1)(P) at photon virtualities 0.006 (GeV/c)(2) < Q(2) < 1 (GeV/c)(2) in the Bjorken x range of 4 x 10(-5) < x < 4 x 10(-2). The results are based on data collected by the COMPASS Collaboration at CERN using muon beam energies of 160 GeV and 200 GeV. The statistical precision is more than tenfold better than that of the previous measurement in this region. In the whole range of x, the measured values of A(1)(p) and g(1)(P) are found to be positive. It is for the first time that spin effects are found at such low values of x. (C) 2018 The Author(s). Published by Elsevier B.V.