The geodynamic environment in which Earth's first continents formed and were stabilized remains controversial. Most exposed continental crust that can be dated back to the Archaean eon (4 billion to 2.5 billion years ago) comprises tonalite-trondhjemite-granodiorite rocks (TTGs) that were formed through partial melting of hydrated low-magnesium basaltic rocks; notably, these TTGs have 'arc-like' signatures of trace elements and thus resemble the continental crust produced in modern subduction settings. In the East Pilbara Terrane, Western Australia, low-magnesium basalts of the Coucal Formation at the base of the Pilbara Supergroup have trace-element compositions that are consistent with these being source rocks for TTGs. These basalts may be the remnants of a thick (more than 35 kilometres thick), ancient (more than 3.5 billion years old) basaltic crust that is predicted to have existed if Archaean mantle temperatures were much hotter than today's. Here, using phase equilibria modelling of the Coucal basalts, we confirm their suitability as TTG 'parents', and suggest that TTGs were produced by around 20 per cent to 30 per cent melting of the Coucal basalts along high geothermal gradients (of more than 700 degrees Celsius per gigapascal). We also analyse the trace-element composition of the Coucal basalts, and propose that these rocks were themselves derived from an earlier generation of high-magnesium basaltic rocks, suggesting that the arc-like signature in Archaean TTGs was inherited from an ancestral source lineage. This protracted, multistage process for the production and stabilization of the first continents-coupled with the high geothermal gradients-is incompatible with modern-style plate tectonics, and favours instead the formation of TTGs near the base of thick, plateau-like basaltic crust. Thus subduction was not required to produce TTGs in the early Archaean eon.