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Context. Along several sight lines within the Milky Way ArH⁺ has been ubiquitously detected with only one detection in extragalactic environments, namely along two sight lines in the redshift z = 0.89 absorber towards the lensed blazar PKS 1830-211. Being formed in predominantly atomic gas by reactions between Ar⁺, which were initially ionised by cosmic rays and molecular hydrogen, ArH⁺ has been shown to be an excellent tracer of atomic gas as well as the impinging cosmic-ray ionisation rates. Aims. In this work, we attempt to extend the observations of ArH⁺ in extragalactic sources to examine its use as a tracer of the atomic interstellar medium (ISM) in these galaxies. Methods. We report the detection of ArH⁺ towards two luminous nearby galaxies, NGC 253 and NGC 4945, and the non-detection towards Arp 220 observed using the SEPIA660 receiver on the APEX 12 m telescope. In addition, the two sidebands of this receiver allowed us to observe the N_KaKc = 1_1,0 - 1_0,1 transitions of another atomic gas tracer p-H₂O⁺ at 607.227 GHz with the ArH⁺ line, simultaneously. We modelled the optically thin spectra of both species and compared their observed line profiles with that of other well-known atomic gas tracers such as OH⁺ and o-H₂O⁺ and diffuse and dense molecular gas tracers HF and CO, respectively. Results. Assuming that the observed absorption from the ArH⁺, OH⁺, and H₂O⁺ molecules are affected by the same flux of cosmic rays, we investigate the properties of the different cloud layers. Based on a steady-state analysis of the chemistry of these three species and using statistical equilibrium calculations, we estimate the molecular fraction traced by ArH⁺ to be ∼ 10^–3 and find that ArH⁺ resides in gas volumes with low electron densities. We further study the ortho-to-para ratio of H₂O⁺ and find that the derived ratios do not significantly deviate from the equilibrium value of three with spin temperatures greater than 15 and 24 K.