The search for a fossil record of Earth's deep biosphere, partly motivated by potential analogies with subsurface habitats on Mars, has uncovered numerous assemblages of inorganic microfilaments and tubules inside ancient pores and fractures. Although these enigmatic objects are morphologically similar to mineralized microorganisms (and some contain organic carbon), they also resemble some abiotic structures. Palaeobiologists have responded to this ambiguity by evaluating problematic filaments against checklists of “biogenicity criteria”. Here, we describe material that tests the limits of this approach. We sampled Jurassic calcite veins formed through subseafloor serpentinization, a water–rock reaction that can fuel the deep biosphere and is known to have occurred widely on Mars. At two localities ~4 km apart, veins contained curving, branched microfilaments composed of Mg-silicate and Fe-oxide minerals. Using a wide range of analytical techniques including synchrotron X-ray microtomography and scanning transmission electron microscopy, we show that these features meet many published criteria for biogenicity and are comparable to fossilized cryptoendolithic fungi or bacteria. However, we argue that abiotic processes driven by serpentinization could account for the same set of lifelike features, and report a chemical garden experiment that supports this view. These filaments are, therefore, most objectively described as dubiofossils, a designation we here defend from criticism and recommend over alternative approaches, but which nevertheless signifies an impasse. Similar impasses can be anticipated in the future exploration of subsurface palaeo-habitats on Earth and Mars. To avoid them, further studies are required in biomimetic geochemical self-organization, microbial taphonomy and micro-analytical techniques, with a focus on subsurface habitats.