I think that you can use splinter regardless of the material. In Merriam-Webster's Unabridged Dictionary online (subscription required), the example sentence for splinter in the meaning you intend references metal.
splin·ter
noun \ˈsplintə(r)\
plural -s
1 a (1) : a thin often jagged
or needlelike piece split or rent off lengthwise : sliver, chip,
fragment
gloves … protect a worker from sharp steel splinters —
Michael Cawley
the steering wheel … was knocked into splinters by an
enemy shot — Edward Breck
(2) : a small jagged or needlelike
particle or flash
flying splinters of ice — William Beebe
irradiated for a moment now and then by splinters shooting through
the darkness — E. K. Brown
And I think sliver is an acceptable synonym for splinter in your use. Here's the definition for sliver from the same source:
sliv·er
noun \ˈslivə(r)\
plural -s
1 a :
a long slender piece cut or torn off : splinter
a piece of apple pie
with a sliver of cheese on top — F. C. Othman
was building up the
fire with split logs and pine slivers — William Faulkner
Neither of these words refer to the action of getting a fragment stick under the skin, so in formal use you would have to specify: "Last time Bob tried whittling, a splinter stuck in his finger, and he couldn't get it out for days." But when the use is informal, I don't think it's a problem to imply the action: "Ow! I just got a splinter!"