procès

The French word procès desperately looks like it should mean the same as English process.

Well, in a sense, it does, but usually to a very specific type of process, namely a trial, lawsuit or legal proceedings. The phrase engager un procès means to take legal action, to file suit (US).

Less literally, procès can also mean an "accusation". So if somebody says "tu me fais un mauvais procès!", they're accusing you of groundless accusations, of misrepresenting their words or actions.

Just occasionally in scientific writing, procès is actually used to mean process. For example, you could talk about a procès biologique ("biological process").

However, in management, technology, politics etc, the normal word for "process" in the sense of "set of events/procedures that must run its course" is processus. For example, a peace process is a processus de paix. The words procédé and méthode are usually used to mean "process" in the sense of "invented method".