The rapid spread of the new coronavirus across the world has raised concerns about the responsiveness of cities and healthcare systems during pandemics. Recent studies try to model how the number of COVID-19 infections will likely grow and impact the demand for hospitalization services at national and regional levels. However, less attention has been paid to the geographic access to COVID-19 healthcare services and to the response capacity of hospitals at the local level, particularly in urban areas in the Global South. This paper shows how transport accessibility analysis can provide actionable information to help improve healthcare coverage and responsiveness. It analyzes accessibility to COVID-19 healthcare at high spatial resolution in the 20 largest cities of Brazil. Using network-distance metrics, we estimate the vulnerable population living in areas with poor access to healthcare facilities that could either screen or hospitalize COVID-19 patients. We then use a new balanced floating catchment area (BFCA) indicator to estimate spatial, income and racial inequalities in access to hospitals with intensive care unit (ICU) beds and mechanical ventilators while taking into account congestion effects. Based on this analysis, we identify substantial social and spatial inequalities in access to health services during the pandemic. The availability of ICU equipment varies considerably between cities and it is substantially lower among black and poor communities. The study maps territorial inequalities in healthcare access and reflects on different policy lessons that can be learned for other countries based on the Brazilian case.