This article proposes a novel all-metal mechanical phase shifter in gap waveguide technology. The phase shifter is aimed at providing beam-scanning capabilities to conventional slot array antennas along the elevation plane. To validate experimentally the beam-steering functionality, a 4×8 slot-array antenna has been designed and fabricated, along with the phase-shifting mechanism. The whole antenna consists of two pieces: a lower rotatable block, which changes the length of concentric groove gap waveguides, and an upper fixed block, where the slot-array antenna is placed. Experimental results validate the proposed concept, having obtained steering angles of up to 25°, with gain levels around 20 dBi with an antenna efficiency close to 90%. A reflection coefficient below -10 dB is achieved for a wide range of rotation angles from 29.5 to 30.5 GHz. The proposed phase shifter is completely scalable to any array size and its true-time-delay nature enables wide steering ranges for closely-spaced slot arrays with wideband radiation performance.