If space-time "sags" under the influence, of massive objects:

 

and if matter "grips" space-time, so as to "curl space-time up", i.e. induce positive "contractile" curvature (Wheeler. Journey into Gravity & Spacetime); then might relativistically-dense black-holes be able to "curl the fabric of space-time all the way around", and "pinch off" (vaguely resembling a rain-drop) ??

 

If space-time "sags" under the influence, of massive objects:

 

and if matter "grips" space-time, so as to "curl space-time up", i.e. induce positive "contractile" curvature (Wheeler. Journey into Gravity & Spacetime); then might relativistically-dense black-holes be able to "curl the fabric of space-time all the way around", and "pinch off" (vaguely resembling a rain-drop) ??

 

I am probably being very stupid but wouldn't this mean it would shut itself down.

This results in a topological change of the space-time, which is generally believed not to be very natural in classical general relativity. Usually topological changes are dynamically constrained, they will not obey the field equations with any sensible source of matter. Geroch has shown that topological change can be possible in GR, but the cases he considers show causality violation. You should have a good hunt through the arXiv for papers on "topology change in general relativity".

 

For quantum gravity the situation is probably different. It is expected that one would have to consider not just fluctuations in the geometry but also the topology.

Quick reply: No. If yes, I think we'd see the effects of such a "pinch" in astronomical observations.