We present a new 300 ks Chandra observation of M87 that limits pileup to only a few per cent of photon events and maps the hot gas properties closer to the nucleus than has previously been possible.Within the supermassive black hole's gravitational sphere of influence, the hot gas is multiphase and spans temperatures from 0.2 to 1 keV. The radiative cooling time of the lowest temperature gas drops to only 0.1-0.5Myr, which is comparable to its free fall time. Whilst the temperature structure is remarkably symmetric about the nucleus, the density gradient is steep in sectors to the N and S, with ρ ∝ r-1.5 ± 0.1, and significantly shallower along the jet axis to the E, where ρ ∝ r-0.93 ± 0.07. The density structure within the Bondi radius is therefore consistent with steady inflows perpendicular to the jet axis and an outflow directed E along the jet axis. By putting limits on the radial flow speed, we rule out Bondi accretion on the scale resolved at the Bondi radius. We show that deprojected spectra extracted within the Bondi radius can be equivalently fitted with only a single cooling flow model, where gas cools from 1.5 keV down below 0.1 keV at a rate of 0.03M⊙ yr-1. For the alternative multitemperature spectral fits, the emission measures for each temperature component are also consistent with a cooling flow model. The lowest temperature and most rapidly cooling gas in M87 is therefore located at the smallest radii at ~ 100 pc and may form a mini cooling flow. If this cooling gas has some angular momentum, it will feed into the cold gas disc around the nucleus, which has a radius of ~ 80 pc and therefore lies just inside the observed transition in the hot gas structure.