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The standard model of cosmology, the LCDM model, robustly predicts the existence of a multitude of dark matter 'subhaloes' around galaxies like the Milky Way. A wide variety of observations have been proposed to look for the gravitational effects such subhaloes would induce in observable matter. Most of these approaches pertain to the stellar or cool gaseous phases of matter. Here we propose a new approach, which is to search for the perturbations that such dark subhaloes would source in the warm/hot circumgalactic medium (CGM) around normal galaxies. With a combination of analytic theory, carefully-controlled high-resolution idealised simulations, and full cosmological hydrodynamical simulations (the ARTEMIS simulations), we calculate the expected signal and how it depends on important physical parameters (subhalo mass, CGM temperature, and relative velocity). We find that dark subhaloes enhance both the local CGM temperature and density and, therefore, also the pressure. For the pressure and density, the fluctuations can vary in magnitude from tens of percent (for subhaloes with M_sub=10^10 Msun) to a few percent (for subhaloes with M_sub=10^8 Msun), although this depends strongly on the CGM temperature. The subhaloes also induce fluctuations in the velocity field ranging in magnitude from a few km/s up to 25 km/s. We propose that X-ray, Sunyaev-Zel'dovich effect, radio dispersion measure, and quasar absorption line observations can be used to measure these fluctuations and place constraints on the abundance and distribution of dark subhaloes, thereby placing constraints on the nature of dark matter.