A new space era is fast approaching. A multitude of miniaturised probes will soon permeate the inner solar system. The abundantly variegated minor bodies will be the destinations of numerous missions driven by exploration and exploitation needs. Missions to rocky planets will feature networks of artificial satellites to support science and operations. Yet, the state-of-the-art is to pilot deep-space probes from ground. Although this is reliable, ground control slots will saturate soon, thus hampering the current momentum in space exploration. EXTREMA enables self-driving spacecraft: machines able to travel in the deep space free of human-driven instructions. We take the challenge to make these systems a reality, and fundamental research is conducted to lay down their foundations. The ambition of EXTREMA is to prove that minor bodies and inner planets can be reached in a totally autonomous fashion with highly constrained platforms. These systems are used to engineer ballistic capture, an extremely rare event observed in highly sensitive regimes. To reinforce this logic, a new approach in orbit validation is introduced, which excels pure computer simulations. Erected over three pillars, the project forges a Simulation Hub, which reproduces on ground the spacecraft-environment interaction. While the pillars enable intermediate milestones, such as inferring the spacecraft position by exploiting the surrounding environment (autonomous navigation), self-determining a nominal plan without a-priori knowledge (autonomous guidance), and targeting the corridors that conduce to ballistic capture, it is the activity performed in the Simulation Hub that allows achieving the objectives via dedicated case studies. A successful outcome will boost access to outer space. The impact is to favour settlements in the inner solar system on a large-scale basis. Located at the fringe of research, EXTREMA can determine a paradigm shift in the way we conceive and conduct deep-space mission.