Exactly. Now draw it out further. Christ is the living Word. Christ is the living Manna. And so Mary is the living Ark. This is not a new analogy either--Christian writers in the first few centuries often made this point.
Is there any direct biblical evidence? My 5th grader is asking what Catholics believe and I would like to be able to answer him ... but I genuinely dont know how Catholics get to Mary being sinless.
When the angel Gabriel salutes Mary he calls her "kecharitomene". We translate that "full of grace", but it really is a Greek passive perfect verb: "the one who has been graced". The Greek usage here indicates an action that was completed and perfected in the past: i.e. Mary received grace so completely and fully that she is no longer in need of any more. That's why "full of grace" is sometimes used to translate it.
That's not exactly proof, but it explains why you see early Christian writers (who knew Greek very well) referring to Mary as "immaculate" and "all-holy".
This idea runs through Christian literature since almost the very beginning--and I am not aware of anyone saying the contrary-: "Mary was a sinful person just like us." The Church, therefore, felt comfortable defining the dogma because no serious challenges to it were made until very late--the time of the Reformation I believe.
Is it possible that she was sinless without her recognizing that she was?