It requires a bit of calculation, but it is possible to show that relativity poses a limit to the tensile strength of any material just enough to prevent such a cylinder from supporting itself
Even if it was so, the cylinder would exist for a while. It would not be crushed instantly.
But the tensile strength needed, is almost arbitrary small.
I guess I should have made my conclusions explicit:
Classically, the escape velocity is independent of the direction of emission, because the gravitational force is potential (unlike, say, magnetism or friction). In GR the situation is more complicated because of the potential capture by an event horizon.
Light always escapes, regardless of direction (assuming your cylinder is transparent), if there is no horizon close by. In other words, the only time a ray of light can be captured is when it dips under the event horizon. This is basically the definiti
As mister shminux mentioned somewhere, he is happy and qualified to answer questions in the field of the Relativity. Here is mine:
A long rod (a cylinder) could have a large escape velocity in the direction of its main axe. From its end, to the "infinity". Larger than the speed of light. While the perpendicular escape velocity is lesser than the speed of light.
Is this rod then an asymmetric black hole?