We didn't keep power for more than 15 minutes -- maximum. We switched off three of the four groups, but it still happened. We disconnected everything from the wall sockets -- still happened. We even went so far as to disconnect the fridge -- still happened.
So, it must be the installation itself. Before we moved in, we did make a modification in the electrical wiring: we added a ground wire and a power-carrying wire from one side of the living room to the other. The conduit runs through the lowered ceiling of the part that had been built later. Especially the second wire took quite a bit of persuasion to get through the conduit.
There are some power-hungry devices who get their power through that line: fridge, oven, dishwasher...
Perhaps the wires got too hot, and the insulation degraded. Perhaps the current-carrying wire leaks current to the grounding wire -- which would certainly explain why not the circuit breakers of the group flip, but the ground fault circuit breaker.
Today, my father-in-law made room in his busy schedule for today to come over and fix the problem. Until that time, we have switched off the offending group, and run our equipment downstairs via a very long extension cable.