Beamed directly into your brain by: unstattedCommoner
The author writes:
I can confirm that I have given sufficient information for the reader to determine (a) whether each of A, B, and C is a knight or a knave, (b) which of "Bal" and "Da" means "yes" and which means "no", (c) who is or is not a member of A's Club, (d) whether each box was made by Bellini, Cellini, a son of Bellini or a son of Cellini, (e) which box contains the cat, and (f) whether the cat is alive or dead. All will be revealed in due course.
Yes, I have recently read Smullyan's What is the name of this book?
EDIT: In response to the forum thread, you also need to know that if a set of inhabitants forms a club, then the complement of that set is also a club. It should be obvious from A's answer in panel 2 that the set of knaves cannot be a club, so the set of knights isn't either.
EDIT 2: Yes, you should assume that Blackdrip (i) is neither a knight nor a knave, (ii) is not an inhabitant of the island, and (iii) is not a member of any club.
EDIT 3: A correspondent has claimed that there are in fact two distinct solutions to the problem. That it true, unless the exchange in panel 3 takes place on a Sunday, in which case the solution is unique. Since I said I had given sufficient information, it must be the case that panel 3 took place on a Sunday.