Small-pitch 3D silicon pixel detectors have been investigated as radiation-hard candidates for the innermost layers of the HL-LHC pixel detector upgrades. Prototype 3D sensors with pixel sizes of 50×50 and 25×100μm2 connected to the existing ATLAS FE-I4 readout chip have been produced by CNM Barcelona. Irradiations up to particle fluences of 3×1016 neq/cm2, beyond the full expected HL-LHC fluences at the end of lifetime, have been carried out at Karlsruhe and CERN. The performance of the 50×50μm2 devices has been measured in the laboratory and beam tests at CERN SPS. A high charge collected and a high hit efficiency of 98% were found up to the highest fluence. The bias voltage to reach the target efficiency of 97% at perpendicular beam incidence was found to be about 100 V at 1.4×1016 neq/cm2 and 150 V at 2.8×1016 neq/cm2, significantly lower than for the previous IBL 3D generation with larger inter-electrode distance and than for planar sensors. The power dissipation at -25°C and 1.4×1016 neq/cm2 was found to be 13 mW/cm2. Hence, 3D pixel detectors demonstrated superior radiation hardness and were chosen as the baseline for the inner layer of the ATLAS HL-LHC pixel detector upgrade.