Two forthcoming rover missions (NASA’s Perseverance rover and the ESA/Roscosmos Rosalind Franklin rover) aim to seek geological biosignatures (fossils, broadly conceived) in rocks formed in habitable environments on early Mars. Promising samples found by Perseverance may later be returned to Earth for further analysis. An important aspect of the search for biosignatures on Mars is the need to exclude “false positive” results. As palaeobiologists know very well, microbial fossils can be difficult to distinguish from misleadingly lifelike geochemical and morphological signals produced by abiotic processes. Here, I will: (1) suggest several ways in which pseudofossils might have formed on early Mars; (2) reinterpret some high-profile purported fossil microbes (including Earth’s supposed oldest) as probable pseudofossils in light of experimental results; (3) present a new assemblage of dubiofossils from a Mars-analogue palaeoenvironment; (4) critically discuss the ways in which palaeobiologists and astrobiologists (including Oxford’s late Martin Brasier) have dealt with the problem of pseudofossils in the past. I will conclude that astrobiologists need to work with physical chemists to explore the range of lifelike objects, patterns and substances that far-from-equilibrium geochemical processes can produce.