Two Citroen-friends did all the work actually, I was not prepared, and I'm having some stiff back problems for some time now. So no credits for me except documenting it here.
In this mode of failure, the suspension pump starts blowing its 40A fuse. Starts with blowing after prolonged use (multiple up-downs for tire changing etc), then starts blowing every time the pump kicks in. Usually a higher amperage fuse seems to "resolve" the issue, which I have seen on C5s with dead pump motors.
What happens is, carbon "dust" from the motor brushes accumulate between copper conductors on the rotor inside. This causes the electric current drawn to rise, up to a point where it starts blowing the 40A fuse. The reason a higher amp fuse seems to resolve is, the current draw might have reached somewhere higher than 40A but lower than 60A. The internals and possibly the wiring is not designed for this, so prolonged use with the higher amp fuse can damage the pump motor and/or wiring irreversibly, meaning expensive repairs.
Considering the newest Mk1 C5 is around 15 years old, I recommend doing this even if you are not experiencing the symptoms and planning to use the car for some more years. The car we worked on is a 2005 Mk1 Phase 2 C5, with the 3.0 v6 engine. Before the work was carried out, the pump motor would blow the 40A fuse everytime it kicked in, meaning it would draw more than 40 Amperes. After the work, we measured less than 15 Amperes *under the same conditions*.
Note that you'll need to get the motor on the bench; so this requires depressurizing the system, removing front right wheel, wheel arch liner, dropping the right side of the bumper, removing the right headlight, and LDS tank. Others have much better instructions than I have for these, so please look for those parts of the process elsewhere.
Once you have the pump on the bench, open the cap using Torx bits (T20 or similar size). You'll see the brush assembly. Remove that too.
Here's how the rotor in my pump motor looked like:
See the "gaps" between the copper conductors?
From top: Give everyghing a thorough clean using an old toothbrush and brake cleaner/IPA (alcohol)/circuit cleaner. While at it, you can separate the pump portion and clean the internals, but that's a separate process which requires its own guide. Also found elsewhere. If you can solder, you can also replace the carbon brushes. On my case, the commutator (the assembly with those copper conductor parts on) had scratches and uneven wear. We leveled those as well with sandpaper, rotating the assembly with a power drill.