sliver

bit, disk (or disc), fragment, part, particle, portion, scrap, section, shard

flinders, shiver, smithereens

shred, tatter

clipping, paring, shave, shaving, snippet

leaf, sheet, slice

chip, chop, dice, hash, julienne, mince

saw, scissor

cleave, rive, split

gash, incise, rip, slash, slit