The only description given of it is that the hero should "shun the frumious Bandersnatch." It is not said that there are not, necessarily, non-frumious bandersnatches, but merely that the hero of the poem must shun a Bandersnatch that is frumious.
In Larry Niven's fictional Known Space universe, the Bandersnatchi are colossal sluglike creatures. They were created by the Tnuctipun[?], used by the Thrintun as genetic engineers, as a food source by the Thrintun. Unknown to the Thrintun, the Bandersnatchi were created sentient, as spies, and immune to telepathy.
wikipedia.org dumped 2003-03-17 with terodump