The TOTEM experiment has made a precise measurement of the elastic proton–proton differential cross-section at the centre-of-mass energy s=8 TeV based on a high-statistics data sample obtained with the β⁎=90 m optics. Both the statistical and systematic uncertainties remain below 1%, except for the t -independent contribution from the overall normalisation. This unprecedented precision allows to exclude a purely exponential differential cross-section in the range of four-momentum transfer squared 0.027<|t|<0.2 GeV2 with a significance greater than 7 σ . Two extended parametrisations, with quadratic and cubic polynomials in the exponent, are shown to be well compatible with the data. Using them for the differential cross-section extrapolation to t=0 , and further applying the optical theorem, yields total cross-section estimates of (101.5±2.1) mb and (101.9±2.1) mb , respectively, in agreement with previous TOTEM measurements.