The first measurement of heavy-flavour production in a fixed-target mode at the LHC is presented. The production of $J/\psi$ and D$^0$ mesons are studied in collisions of protons with energies of $4\mathrm{\,Te\kern -0.1em V}$ and $6.5\mathrm{\,Te\kern -0.1em V}$ incident on helium and argon nuclei at rest at centre-of-mass energies of $\sqrt{s_{nn}}~=~86.6\mathrm{\,Ge\kern -0.1em V}$ and $\sqrt{s_{nn}}~=~110.4\mathrm{\,Ge\kern -0.1em V}$, respectively.