The brown pudding's attacks do no damage, and its AC is rather high; its only danger lies in rotting your or your pet's armor. This can be easily avoided by any combination of wearing metal armor, covering your body armor with a throwaway cloak, or simply removing any organic armor. Pets without armor, such as domestic animals like dog, can freely dispose of a brown pudding.
If you have a pudding pet, you can hit it with iron items to cause the pet to split, yielding more pets for polymorphing; see abuse for the ramifications of attacking and splitting your pet. As the name is cloned, you might want to rename one. Alternately, consider splitting them first, then using a scroll of taming or a spell of charm monster if either is available. For pet splitting, a Puddingbane is recommended to avoid incurring alignment and luck penalties.
Beginning in NetHack 3.6.0, brown puddings leave globs instead of corpses when they are killed. Globs cannot be sacrificed, revived or tinned, making pudding farming a less useful tactic than in previous versions of NetHack. Globs also could not give resistances when eaten; this is fixed as of 3.6.1.
"Blob", "pudding", and "ooze" share the same entry:
These giant amoeboid creatures look like nothing more than
puddles of slime, but they both live and move, feeding on
metal or wood as well as the occasional dungeon explorer to
supplement their diet.
But we were not on a station platform. We were on the track ahead
as the nightmare, plastic column of fetid black iridescence oozed
tightly onward through its fifteen-foot sinus, gathering unholy
speed and driving before it a spiral, re-thickening cloud of the
pallid abyss vapor. It was a terrible, indescribable thing vaster
than any subway train -- a shapeless congeries of protoplasmic
bubbles, faintly self-luminous, and with myriads of temporary eyes
forming and unforming as pustules of greenish light all over the
tunnel-filling front that bore down upon us, crushing the frantic
penguins and slithering over the glistening floor that it and its
kind had swept so evilly free of all litter.