The Bandersnatch is a fictional character|fictional creature mentioned in Lewis Carroll's poems Jabberwocky and The Hunting of the Snark. The form or size of the creature is not described. Nor is it clear whether the Bandersnatch is a singularity, like the Phoenix.
The only description given of it is that the hero should "shun the frumious Bandersnatch." Other advice given to the beamish hero proving quite accurate, one must dismiss the possibility that the speaker within the poem was uninformed about the Bandersnatch, which, however, does not actually make an appearance in "Jabberwocky". Thus it is clear that the Bandersnatch is not of a size or character to be embraced. Elsewhere in Through the Looking-Glass, however, it is implied (but not stated) that a Bandersnatch is quick-moving, and that there may be more than one of it: "She runs so fearfully quick. You might as well try to catch a Bandersnatch!"
It is not stated whether there are non-frumious Bandersnatches, or whether these can be approached safely, merely that the hero of the poem must shun a Bandersnatch that is frumious. At the same time, it may be that 'frumious' is not a merely descriptive adjective, but a definitive one, describing the essential quality of the Bandersnatch. 'Frumious' is a portmanteau word combining 'furious' and 'fuming'.
Fortunately, Carroll's The Hunting of the Snark (Fit the Seventh) settles some of these questions: in an encounter with a bandersnatch it is described as moving swiftly, having a neck it can extend, and having snapping, frumious jaws, with which it tries to grab the Banker.
Further developments
At least two authors other than Carroll have addressed the subject of bandersnatches.
New Alice
The book A New Alice in the Old Wonderland, 1895, by Anna M. Richards, contains a broader description given of the Bandersnatch with the poem Bandersnatchy.
In this poem another hero sets out to slay the frumious Bandersnatch so as to gain respect from his people against the hero who slew the Jabberwock (a story he would sit and tell till after ten o'clock).
It is necessary to be armed with a vorpal sword or a winxy pistol, because one never can tell what a Bandersnatch might do. According to the hero, the Bandersnatch is a queer thing that whizzles overhead causing him to believe it is up a tree, only to his surprise its legs are awfully high. It pleads the hero not to shoot only to fool him and fly away. The hero grabs its tail and cuts it off as his trophy.
The hero describes the creature as being extremely long legged with a long tail as well and the ability to fly extremely fast. It could be understood that the Bandersnatch perhaps camouflages itself as a tree, given the explanation understood by the hero's description.
There is also an added illustration by Anna M. Richards of the hero's encounter with the Bandersnatch.
Omar
The book Omar: A fantasy for animal lovers by Wilfrid Blunt (ASIN B0006BVCT6) states that bandersnatch is in fact a rock hyrax, which is only dangerous when it is frumious.
Larry Niven
In Larry Niven's Known Space series, the Bandersnatch (Known Space) are a race of aliens found in several parts of the galaxy, but especially on the planet Jinx. They are an engineered species of enormous intelligent single-celled animals.