Gamma‐ray bursts (GRBs) are powerful probes of the early Universe, but locating and identifying very distant GRBs remain challenging. We report here the discovery of the K‐band afterglow of Swift GRB 060923A, imaged within the first hour post‐burst, and the faintest so far found. It was not detected in any bluer bands to deep limits, making it a candidate very high‐z burst (z≳ 11). However, our later‐time optical imaging and spectroscopy reveal a faint galaxy coincident with the GRB position which, if it is the host, implies a more moderate redshift (most likely z≲ 2.8) and therefore that dust is the likely cause of the very red‐afterglow colour. This being the case, it is one of the few instances so far found of a GRB afterglow with high‐dust extinction.