Actually your figure is correct. I messed up the calculation.
I get 3,096 fps (v2=2Gm/r).
NASA says 1.022 kilometers per second.
Good job.
Lots of small arms fire projectiles at above 1 K/S, including the AR-15. Of course, those small objects would not make it through the atmosphere intact.
But bigger objects could have heat shields and guidance systems.
We already have 155mm smart artillery shells that travel close to that.