Is it possible to stall a plane so badly that the nose refuses to go down due to lack of airspeed?
Yes it's called a Deep Stall, and is mostly a problem with T Tail aircraft, especially jets with Supercritical Airfoil wings (like the CRJ Regional Jet line).  

Such wings stall from the leading edge and the stall's flow separation spreads rapidly and completely across the entire wing all at once, so there is very little residual nose down pitching moment.  Plus the T tail ends up in a spot where it's in the wake of the flow from the wing, and gets blanked out, and loses its ability to pitch the nose over with positive lift since it's sitting there in the turbulent wake of the wing.  

So the plane just mushes downward in an unrecoverable, stabilized, well, mush, and will pancake into the ground like that.  A CRJ200 test aircraft was lost in development testing when it got into a deep stall and IIRC, the stall/spin recovery parachute in the tail failed to deploy (or it didn't have one at the time; I forget which).

Such airplanes require, in addition to the usual stick shaker, a stick pusher to force the nose over, by shoving the control column for you if you do nothing while the shaker is going off, before the natural stall can occur, since the natural stall can be unrecoverable.  Generally if an airplane has a stick pusher system it means it has a deep stall mode.