I am totally late for this party, but in case you still care... In "Objects in Space" Shepherd Book is shown coming out of what is presumably a bathroom in the common area outside the infirmary. It might contain a shower... or it might not. The diagram of the ship in the Serenity RPG (which is not technically canon but can be handy anyway) does show a shower in there. The crew presumably have sinks and toilets in each of their rooms, based on the shot we got of Mal's bunk in the pilot.
Personally, I like to think that room by the passenger dorms has a shower. I shudder to think of all those folks cooped up on that ship for weeks with nothing but spit baths. And I can't really imagine that Inara could do business that way.
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I am totally late for this party, but in case you still care... In "Objects in Space" Shepherd Book is shown coming out of what is presumably a bathroom in the common area outside the infirmary. It might contain a shower... or it might not. The diagram of the ship in the Serenity RPG (which is not technically canon but can be handy anyway) does show a shower in there. The crew presumably have sinks and toilets in each of their rooms, based on the shot we got of Mal's bunk in the pilot.
Personally, I like to think that room by the passenger dorms has a shower. I shudder to think of all those folks cooped up on that ship for weeks with nothing but spit baths. And I can't really imagine that Inara could do business that way.