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 fourmomentum transfer squared 0.027 < |t| < 0.2 GeV 2 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.This article is dedicated to the memory of Prof. E. Lippmaa and Prof. M. Lo Vetere who passed away recently