We report results from a new analysis of a deep 95-μm imaging survey with the photopolarimeter ISOPHOT on-board the Infrared Space Observatory, over a 40 × 40 arcmin2 area within the Lockman Hole. To this end we exploit a newly developed parametric algorithm able to identify and clean spurious signals induced by cosmic rays impacts and by transient effects and non-linearities in the detectors. These results provide us with the currently deepest - to our knowledge - far-infrared (far-IR) image of the extragalactic sky. Within the survey area, we detect 36 sources with signal-to-noise ratio S/N > 3 (corresponding to a flux of 16 mJy), making up a complete flux-limited sample for S95μm>= 100 mJy. Reliable sources are detected, with decreasing but well-controlled completeness, down to S95μm~= 20 mJy. The source extraction process and the completeness, the photometric and astrometric accuracies of this catalogue have been tested by us with extensive simulations accounting for all the details of the procedure. We estimate source counts down to a flux of ~30 mJy, at which limit we evaluate that 10-20 per cent of the cosmic IR background (CIRB) has been resolved into sources (contributing to the CIRB intensity ~=2.0 × 10-9 W m-2 sr-1). The 95-μm galaxy counts reveal a steep slope at S95μm<= 100 mJy (α~= 1.6), in excess of that expected for a non-evolving source population. The shape of these counts agrees with those determined by ISO at 15 and 175 μm, and starts setting strong constraints on the evolution models for the far-IR galaxy populations.