The detection of life beyond Earth is an ongoing scientific endeavour, with profound implications. One approach, known as the search for extraterrestrial intelligence (SETI), seeks to find engineered signals (`technosignatures') that indicate the existence technologically-capable life beyond Earth. Here, we report on the detection of a narrowband signal-of-interest at ~982 MHz, recorded during observations toward Proxima Centauri with the Parkes Murriyang radio telescope. This signal, `BLC1', has characteristics broadly consistent with hypothesized technosignatures and is one of the most compelling candidates to date. Analysis of BLC1 -- which we ultimately attribute to being an unusual but locally-generated form of interference -- is provided in a companion paper (Sheikh et al., 2021). Nevertheless, our observations of Proxima Centauri are the most sensitive search for radio technosignatures ever undertaken on a star target.