We report the spectroscopic confirmation and fundamental properties of TOI$-$757 b, a mini$-$Neptune on a 17.5$-$d orbit transiting a bright star ($V\, =\, 9.7$ mag) discovered by the TESS mission. We acquired high$-$precision radial velocity measurements with the HARPS, ESPRESSO, and PFS spectrographs to confirm the planet detection and determine its mass. We also acquired space$-$borne transit photometry with the CHEOPS space telescope to place stronger constraints on the planet radius, supported with ground$-$based LCOGT photometry. WASP and KELT photometry were used to help constrain the stellar rotation period. We also determined the fundamental parameters of the host star. We find that TOI$-$757 b has a radius of $R_{\mathrm{p}} = 2.5 \pm 0.1 R_{\oplus }$ and a mass of $M_{\mathrm{p}} = 10.5^{+2.2}_{-2.1} M_{\oplus }$, implying a bulk density of $\rho _{\text{p}} = 3.6 \pm 0.8$ g cm$^{-3}$. Our internal composition modelling was unable to constrain the composition of TOI$-$757 b, highlighting the importance of atmospheric observations for the system. We also find the planet to be highly eccentric with e = 0.39$^{+0.08}_{-0.07}$, making it one of the very few highly eccentric planets among precisely characterized mini$-$Neptunes. Based on comparisons to other similar eccentric systems, we find a likely scenario for TOI$-$757 b's formation to be high eccentricity migration due to a distant outer companion. We additionally propose the possibility of a more intrinsic explanation for the high eccentricity due to star$-$star interactions during the earlier epoch of the Galactic disc formation, given the low metallicity and older age of TOI$-$757.