It has been proven that time far away from Earth is faster than time on the surface of Earth, due to gravitational time dilation. (GPS satellites take gravitational time dilation to account.)
Would this mean that the speed of light far from Earth, if measured dividing the distance elapsed by the time elapsed on Earth, would yield a value exceeding $c$, assuming theoretically hypersensitive instruments?
Yes, the speed of a distant beam of light in "flatter" spacetime, as measured using Earth's coordinates, may be very slightly greater than $c$. Special relativity holds only in flat spacetime, or in a small enough region of spacetime that appears to be flat. So if you measure the speed of any beam of light that's passing by you, you'll get $c$, but this need not be the case for distant light in a different region of curved spacetime.
Imagine that you drilled a tunnel through the planet Venus, and then put someone with a laser pointer on Mercury. If he flashed it at Earth, and then when Venues passed in between he flashed it through the tunnel at Earth, then yes it would take longer for the light to hit Earth while passing through the tunnel because of the gravitational time dilation on Venus.
So yes, you are correct in your assumption. But keep in mind that time dilation on Earth is extremely small when compared to distant open space. Compared to open space we find a time dilation factor on Earth of tE=1.000,000,000,699,68 or just 21 centimeters per second difference in the speed of light. You would be much better off by measuring compared to the sun at tS =1.000,002,121,041,69, or to a very large star with a time dilation factor of tR =1.000,022,265,004,83 which is 6,675 meters per second difference in the speed of light.
I find it very helpful to think of the speed of light as a proxy for the speed of time.
The speed of light is always $c$ when measured locally. In Special Relativity, which deals with flat spacetime, then local can be extended to as large a region as you like, but if you introduce gravitating objects that curve spacetime then local and non-local measurements of the light speed will differ. That is because the spacetime coordinates you are using (and indeed any coordinate system) to measure events locally cannot be applied without distortion to other or larger regions of spacetime.
We can demonstrate the varying speed of light in curved spacetime by setting up a race between two photons and have one reach the finish sooner, despite traveling equal distances.
First we need two towers some kilometers apart and of the same height and parallel to each other. We place a light source at the middle of one tower and simultaneously send signals up and down that bounce off mirrors at the top and bottom and return to the centre. We can use an interferometer to determine that the signals return at exactly the same time and that the source is exactly half way up. We do the same to determine the exact mid point on the the other tower. We also use radar measurements by sending a signal from the base of one tower to the base of the other and determine the distance using an accurate atomic clock also at the base of the tower. We repeat the process at the top of the tower using an atomic clock at the top of the tower to ensure that the tops are exactly the same distance apart as the bases.
Now we set up suitable mirrors as per the diagram and start the race. The photon going via the high route arrives first because it is travelling faster at high altitude than the other photon. We could use rulers instead of atomic clocks to ensure the tops are same distance apart as the bases and the result would be the same.
While the above thought experiment demonstrates that the speed of light depends on gravitational potential, can we have a speed of light that really is greater than c? Firstly, the local speed of light is always measured to be c (in a vacuum). Secondly, if we define c as the speed of light in a vacuum very far from any gravitational source, then any gravitational field will only slow the light down rather than speed it up. However an observer on the surface of an extreme gravitational body might measure the velocity of a photon orbiting very high up as being greater than c, but that is only because they are effectively comparing the speed of the high altitude photon to their local speed of light, (which they measure to be c).
For the eagle eyed and observant, the source is not at the geometrical centre of the tower and the lower leg is shorter than than the upper. This is due to gravitational length contraction and together with speed of light varying with the speed of light the optical radar center is not not at the geometrical centre in a gravitational field.
