We discuss transport of different alkali-metal ions in ultra-narrow boron nitride nanotubes (BNNTs), based on experiments with a new, macroscopic experimental platform consisting of vertically aligned nanotubes at densities up to 106 pores/cm2. Our study reveals that ion transport in BNNTs is fundamentally different from that in either carbon nanotubes of similar size, or 2D boron-nitride nanochannels. We find two unexpected transport phenomena: (i) ultra-fast, cation-selective diffusion that exceeds Fickian diffusion by up to 31-fold and (ii) preferentially enhanced transport rates for Li+ over other alkali-metal ions (K+ and Na+) that are opposite to the ordering of their bulk mobilities in solution. We seek to explain these unexpected ion-transport phenomena with a combination of ab initio calculations, molecular-dynamics simulations, and continuum analysis. As a result of enhanced and cation-selective transport, the BNNT membranes produced per-pore osmotic-power densities up 15,300 W/m2 in a LiCl-concentration gradient, with energy-conversion efficiencies approaching the theoretical maximum of 50%. We also discuss our attempts to combine ion selectivity and hydrodynamic flow enhancement by externally coating single-walled carbon nanotubes with a few layers of hexagonal boron nitride. As a demonstration, we power a calculator, watch, and light-emitting diode using 1-cm2 BNNT membranes in a salinity gradient. The unusual transport phenomena in BNNTs, as well as the flexible and scalable membrane-fabrication process, may enable ion-selective nanotube membranes optimized for lithium recovery, “blue” osmotic energy, and other separation and energy-conversion processes.