It turns out not only is this possible, the equipment to make it happen is rather simple.
This video explains in detail how it’s done. To see the effect of a supersonic ball contacting a ping-pong paddle, skip to 5:50.
What? No super slow-motion replay? I’m disappointed!
That’s pretty amazing, actually. You can’t even see the paddle disappearing.
So…who else is planning to try this at home? 
That was my first thought - you can’t show something like that and then say “don’t try this at home”!
I could probably cobble something like that together in my garage over a weekend.
That’s pretty much what happened to the space shuttle Columbia. Or at least that’s what they figured out in post-disaster testing, when they fired a chunk of foam toward a wing panel at about 500 MPH. “Oh, it’s so soft and fluffy, it could never…<BOOM>…HOLY SHIT!”
Good thing they didn’t try this.