Ground → ship Wi-Fi bandwidth in my fast moving spaceship

If you are in a perfectly circular orbit, then your received signals are affected by the relativistic transverse Doppler effect.

The signals you would receive would be blueshifted by the Lorentz factor of the orbital speed. That is the information transferal rate would (potentially) go up (if you can tune to the new frequency).

If you are in a Keplerian orbit, then (in a Newtonian approximation) $$ \frac{mv^2}{r} = G\frac{Mm}{r^2}$$ $$\frac{v}{c} =\sqrt{\frac{GM}{rc^2}} = \sqrt{\frac{r_s}{2r}},$$ where $r_s = 2GM/c^2$.

The transverse Dopper blueshift is by a factor of $$ f_r = f_t\gamma = f_t\left(1 - \frac{v^2}{c^2}\right)^{-1/2} = f_t\left(1 - \frac{r_s}{2r}\right)^{-1/2},$$ where $f_r$ and $f_t$ are the received and transmitted frequencies - assuming that the transmitter is at $r=0$ or at least stationary with respect to the orbital speed.

However, in General Relativity there is also a gravitational redshift given by $$ f_r = f_t \left( \frac{1 - r_s/r}{1 - r_s/r_t}\right)^{1/2},$$ where $r_t$ is the position of the transmitter (which has to be at $r_t > r_s$) and this may modify the conclusion.

Which will win out depends on the radius of the orbit compared with the radius of any "surface" you put your transmitter on.

There are two effects due to relativity. The motion of the spaceship around the satellite results in a time dilation that can be calculated using special relativity.

If the satellite has significant mass (e.g., enough to keep the spaceship in orbit), then general relativity (gravity) plays a role.

From the spaceship's perspective, the two effects are opposite, but not equal in magnitude.

This Wikipedia article re error correction of GPS signals may help.

Per the suggestion by @ChappoSaysReinstateMonica, the following is included here as an easy way to think about it:

Assuming the satellite has essentially no mass, so the spaceship needs to use its engines to keep going in a circular orbit around the satellite, then only special relativity applies. The spacecraft will perceive its received data rate as being a bit faster than the transmitter's data rate on the satellite. A way to know this is via the fact that a clock taken from the satellite to the spacecraft, then after a while brought back to the satellite, will lose a bit of time relative to a clock left on the satellite, but uploaded data is like clock ticks on the satellite.

Doppler effect (relativistic) of the Electromagnetic Waves will take place. As a result you will receive the emitted Wifi signal (EM wave) as squeezed along the time axis (increases frequency). Naturally the WiFi demodulator available with you would report invalid (unrecognised) signal because digital data is encoded on the carrier wave by some form of modulation like Frequency Shift Keying , Phase shift keying etc.

Let's say it's transmitted by FSK. In FSK, the 0s and 1s are transmitted through distinct changes in Carrier Wave frequency held for a particular duration. First of all the EM wave frequency has changed hence your demodulator won't even recognise the EM wave as Wifi signal.

Assume that you have accounted for Doppler effect EM wave frequency change and distinct change in the changes of the changed frequency (obtained from it's corresponding changed Fourier transform) for zero and one encoding, into your custom demodulator. So the EM wave is now recognised as Wifi signal. But as stated above the distinct changes in EM wave frequency is held for a pre agreed time (between transmitter and receiver) to encode 0 & 1. So due to time dilation this agreement breaks. So your demodulator although recognises Wifi signal presence, reports invalid protocol.

Now you might be tempted to think let's account for the time dilation into the demodulator. But that isn't really a good idea because the above stated pre agreed duration was statistically obtained to ensure noise free transmission. So if you try to account for that in your demodulator, you will occasionally receive corrupted data due to noise in the transmission. But if you are okay with it, then sure download speed increases!!!!