The fact that you put "speed" in scare-quotes shows that, even if you don't realize it, you intuitively grasp that the concept you are trying to use doesn't work the way you are saying it does.
Normal objects in free fall, i.e., moving only under the influence of "gravity", move on straight lines in spacetime, just like light does. (They are different straight lines because normal objects have mass while light is massless. But they're still straight lines in both cases.) The apparent "curvature" comes from the curvature of spacetime, not any change in the behavior of the objects. The objects have no way of telling locally whether the spacetime they are in is flat or curved.
> the speed of light in vaccume locally at that point won't be affected?
No. You can't tell anything about spacetime curvature or gravity by local measurements of the speed of light in vacuum.
The fact that you put "speed" in scare-quotes shows that, even if you don't realize it, you intuitively grasp that the concept you are trying to use doesn't work the way you are saying it does.
Normal objects in free fall, i.e., moving only under the influence of "gravity", move on straight lines in spacetime, just like light does. (They are different straight lines because normal objects have mass while light is massless. But they're still straight lines in both cases.) The apparent "curvature" comes from the curvature of spacetime, not any change in the behavior of the objects. The objects have no way of telling locally whether the spacetime they are in is flat or curved.
> the speed of light in vaccume locally at that point won't be affected?
No. You can't tell anything about spacetime curvature or gravity by local measurements of the speed of light in vacuum.