The first measurement of $\phi(1020)$ meson production in fixed-target $p$Ne collisions at $\sqrt{s_{NN}}=68.5$ GeV is presented. The $\phi(1020)$ mesons are reconstructed in their $K^{+}K^{-}$ decay in a data sample consisting of proton collisions on neon nuclei at rest, corresponding to an integrated luminosity of $21.7 \pm 1.4$ nb$^{-1}$, collected by the LHCb detector at CERN. The $\phi(1020)$ production cross-section in the centre-of-mass rapidity range of $-1.8 < y^*<0$ and transverse momentum range of $800 < p_T <6500$ MeV/c is found to be $\sigma=182.7\pm2.7~\text{(stat.)}\pm14.1~\text{(syst)}$ $\mu b$/nucleon. A double-differential measurement of the cross-section is also provided in four regions of rapidity and six regions of transverse momentum of the $\phi(1020)$ meson and compared with the predictions from Pythia and EPOS4, which are found to underestimate the experimental values.