Context. Hydrogen deuteride (HD) rotational line emission can provide reliable protoplanetary disc gas mass measurements, but this molecule is difficult to observe and detections have been limited to three T Tauri discs. No new data have been available since the Herschel Space Observatory mission ended in 2013. Aims. We set out to obtain new disc gas mass constraints by analysing upper limits on HD 1–0 emission in Herschel/PACS archival data from the DIGIT key programme. Methods. With a focus on the Herbig Ae/Be discs, whose stars are more luminous than T Tauris, we determined upper limits for HD in data previously analysed for its line detections. We studied the significance of these limits with a grid of models run with the DALI physical-chemical code, customised to include deuterium chemistry. Results. Nearly all the discs are constrained to M_gas ≤ 0.1 M_⊙, ruling out global gravitational instability. A strong constraint is obtained for the HD 163296 disc mass, M_gas ≤ 0.067 M_⊙, implying Δ_g/d ≤ 100. This HD-based mass limit is towards the low end of CO-based mass estimates for the disc, highlighting the large uncertainty in using only CO and suggesting that gas-phase CO depletion in HD 163296 is at most a factor of a few. The M_gas limits for HD 163296 and HD 100546, both bright discs with massive candidate protoplanetary systems, suggest disc-to-planet mass conversion efficiencies of M_p/(M_gas + M_p) ≈ 10–40% for present-day values. Near-future observations with SOFIA/HIRMES will be able to detect HD in the brightest Herbig Ae/Be discs within 150 pc with ≈ 10 h integration time.