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Estimates of the real death toll of the COVID-19 pandemic have proven to be problematic in many countries, Italy being no exception. Mortality estimates at the local level are even more uncertain as they require stringent conditions, such as granularity and accuracy of the data at hand, which are rarely met. The "official" approach adopted by public institutions to estimate the "excess mortality" during the pandemic draws on a comparison between observed all-cause mortality data for 2020 and averages of mortality figures in the past years for the same period. In this paper, we apply the recently developed machine learning control method to build a more realistic counterfactual scenario of mortality in the absence of COVID-19. We demonstrate that supervised machine learning techniques outperform the official method by substantially improving the prediction accuracy of the local mortality in "ordinary" years, especially in small- and medium-sized municipalities. We then apply the best-performing algorithms to derive estimates of local excess mortality for the period between February and September 2020. Such estimates allow us to provide insights about the demographic evolution of the first wave of the pandemic throughout the country. To help improve diagnostic and monitoring efforts, our dataset is freely available to the research community. [ABSTRACT FROM AUTHOR]