Quantifying the physical mechanisms responsible for the transport of sediments, nutrients and pollutants in the abyssal sea is a long-standing problem, with internal waves regularly invoked as the relevant mechanism for particle advection near the sea bottom. This study focuses on internal-wave-induced particle transport in the vicinity of (almost) vertical walls. We report a series of laboratory experiments revealing that particles sinking slowly through a monochromatic internal wave beam experience significant horizontal advection. Extending the theoretical analysis by Beckebanze et al. (J. Fluid Mech., vol. 841, 2018, pp. 614–635), we attribute the observed particle advection to a peculiar and previously unrecognized streaming mechanism in the stratified boundary layer originating at the lateral walls. This vertical boundary layer streaming mechanism is most efficient for significantly inclined wave beams, when vertical and horizontal velocity components are of comparable magnitude. We find good agreement between our theoretical prediction and experimental results.