Is the speed of light in space a constant or a variable?

Constant, simply because there are almost no partricles to disturb.

then light is unaffected by gravity?

No, light is affected by gravity, but not in its speed. General Relativity (our best guess as to how the Universe works) gives two effects of gravity on light. It can bend light (which includes effects such as gravitational lensing), and it can change the energy of light. But it changes the energy by shifting the frequency of the light (gravitational redshift) not by changing light speed. Gravity bends light by warping space so that what the light beam sees as "straight" is not straight to an outside observer. The speed of light is still constant.

Let us boil it down to the very basics.

C is an aggregate velocity whereby electromagnetic radiation transmits through space.  Considered by differing rationale. we see space in three primary variants (as we discusas propagation of EMR): Minkowski Space, de Sitter Space, and Anti de Sitter Space.  It is the same thing but the physical properties of that same thing considered by different applications.  The mathematical rationale is not that difficult but considering this crowd let's avoid it.

Space "rips and re-weaves" very easily.  Apply a field and it will go in that direction until it redshifts to DC.  That is a greatly oversimplified basis of Minkowski Space.

Space "opens and conducts" energetic particles.  That is an oversimplicication of de Sitter Space.

Space "massively conducts wave-energy" like waves in a pool of water.  That is an oversimplification of Anti de Sitter Space.

Now, what happens when you grab a very large heavy thing and try to shake it around?  Inertia (mass) slows it down but keeps it moving also (Newton's First Law).  If you grab a very small and much lighter thing, and shake it around, you can do so much faster.  (Again, Newton's First Law).

So with particles in space.  The higher the energy, the more resistance to its travel through space, but the farther it will go despite that resistance.

I was just asking to see if it was possible for light to travel at a speed of, say, 11 million miles per second. 

A constant. It's everything else that changes!