We present deep ALMA Band 3 observations of the HCN, HCO⁺, and HNC(4-3) emission in SDP.81, a well-studied z = 3.042; strongly lensed galaxy. These lines trace the high-density gas, which remains almost entirely unexplored in z ≥ 1 galaxies. Additionally, these dense-gas tracers are potentially powerful diagnostics of the mechanical heating of the interstellar medium. While the HCN(4-3) and HNC(4-3) lines are not detected, the HCO⁺(4-3) emission is clearly detected and resolved. This is the third detection of this line in a high-redshift star-forming galaxy. We find an unusually high HCO⁺/HCN intensity ratio of ≥2.2. Based on the modelling of the photodissociation region, the most likely explanation for the elevated HCO⁺/HCN ratio is that SDP.81 has low mechanical heating, making up less than 10% of the total energy budget, along with a sub-solar metallicity of Z ≈ 0.5 Z_⊙. While such conditions might not be representative of the general population of high-redshift dusty galaxies, a lower-than-solar metallicity might significantly impact gas masses inferred from CO observations. In addition, we report the detection of CO(0-1) absorption from the foreground lensing galaxy and CO(1-0) emission from a massive companion to the lensing galaxy, approximately 50 kpc to the south-east.