New experimental results on turbulent boundary layers (TBLs) with wall suction show that it is possible to experimentally realise a turbulent asymptotic suction boundary layer (TASBL), i.e. a boundary layer which becomes independent of the streamwise location and with the suction rate as the only control parameter. Turbulent asymptotic suction boundary layers show a mean-velocity profile with a large logarithmic region and without a clear wake region. If outer-scaling is adopted, using the free stream velocity and the boundary layer thickness as characteristic velocity and length scale, respectively, a single log-law describes the logarithmic region of all the measured TASBLs independently from the suction rate. Streamwise velocity profiles were measured with different hot-wire probe sizes, in order to account for and correct for probe-filtering effects. It emerges that wall suction is responsible for strong damping of the velocity fluctuations, with a decrease of the near-wall peak of the velocity-variance profile ranging from $50\,\%$ to $65\,\%$ when compared with a canonical zero-pressure gradient TBL at comparable Reynolds number. The analysis of the power spectral density maps suggests that the decrease in the turbulent activity can be explained by increased stability of the near-wall streaks.