AX J1754.2−2754, 1RXS J171824.2−402934 and 1RXH J173523.7−354013 are three persistent neutron star low-mass X-ray binaries that display a 2-10 keV accretion luminosity L X of only (1-10) × 1034 erg s−1 (i.e. only 0.005-0.05 per cent of the Eddington limit). The phenomenology of accreting neutron stars which accrete at such low accretion rates is not yet well known and the reason why they have such low accretion rates is also not clear. Therefore, we have obtained XMM-Newton data of these three sources and here we report our analysis of the high-quality X-ray spectra we have obtained for them. We find that AX J1754.2−2754 has L X ∼ 1035 erg s−1, while the other two have X-ray luminosities about an order of magnitude lower. However, all sources have a similar, relatively soft, spectrum with a photon index of 2.3-2.5, when the spectrum is fitted with an absorbed power-law model. This model fits the data of AX J1754.2−2754 adequately, but it cannot fit the data obtained for 1RXS J171824.2−402934 and 1RXH J173523.7−354013. For those sources, a clear soft thermal component is needed to fit their spectra. This soft component contributes 40-50 per cent to the 0.5-10 keV flux of the sources. The presence of this soft component might be the reason why the spectra of these two sources are soft. When including this additional spectral component, the power-law photon indices are significantly lower. It can be excluded that a similar component with similar contributions to the 2-10 keV X-ray flux is present for AX J1754.2−2754, indicating that the soft spectrum of this source is mostly due to the fact that the power-law component itself is not hard. We note that we cannot exclude that a weaker soft component is present in the spectrum of this source which only contributes up to ∼25 per cent to the 0.5-10 keV X-ray flux. We discuss our results in the context of what is known of accreting neutron stars at very low accretion rate.