The gravitationally lensed quasar J014516.6-094517 at z = 2.719 has been observed with the ESPRESSO instrument at the ESO VLT to obtain high-fidelity spectra of the two images A and B with a resolving power R = 70 000. At the redshifts under investigation (2.1 ≲ z ≲ 2.7), the Lyman forests along the two sightlines are separated by sub-kiloparsec physical distances and exhibit a strong correlation. We find that the two forests are indistinguishable at the present level of signal-to-noise ratio and do not show any global velocity shift, with the cross-correlation peaking at Δv = 12 ± 48 $\rm m~s^{-1}$. The distribution of the difference in velocity of individual Ly α features is compatible with a null average and a mean absolute deviation of 930 $\rm m~s^{-1}$. Significant differences in NH I column density are not detected, putting a limit to the RMS fluctuation in the baryon density on ≲1 proper kpc scales of Δρ/ρ ≲ 3 per cent. On the other hand, metal lines show significant differences both in velocity structure and in column density. A toy model shows that the difference in velocity of the metal features between the two sightlines is compatible with the motions of the baryonic component associated with dark matter haloes of typical mass M ≃ 2 × 1010 M⊙, also compatible with the observed incidence of the metal systems. The present observations confirm the feasibility of the Sandage test of the cosmic redshift drift with high-fidelity spectroscopy of the Lyman forest of distant, bright quasars, but also provide an element of caution about the intrinsic noise associated with the usage of metal features for the same purpose.