Is gm (and thus device fT) independent of overdrive voltage (Eq. [9.57]) 
in nanometer (or long-channel) CMOS?

Both gm and fT vary greatly with the overdrive voltage. See Eq. (9.55) 
(or, for long-channels, Eq. [9.36]) and Fig. 9.33. 

The most important aspect of this dependence is that to increase CMOS 
device speeds (fT) you need to increase the overdrive voltage. This is
one of the reasons why we bias with the BMR (it's easy to set overdrive
voltage with changes in temperature and process).

The confusion comes from claims that short channel devices' fT
does not vary with overdrive voltage because of saturation velocity effects
(as seen in Eq. (9.57) if vsat is constant). Vsat isn't constant and does
depend on both VGS and VDS as indicated on page 299 (and clearly gm does 
increase with overdrive as seen in Fig. 9.33). The reasons for the 
variations in Vsat are outside the scope of the book (e.g. velocity
overshoot).

Biasing is often a neglected topic when learning to design CMOS analog ICs.
This is unfortunate because it is the MOST IMPORTANT topic in analog IC 
design. When I look at someone else's design the first things I look at 
are the threshold voltage of the process and then VGS (so I can determine 
overdrive). If the VGS is close to the threshold voltage (so overdrive 
is close to zero) then I know that the design will be inherently slow (and
operating in weak inversion). 

If the overdrive voltages vary from device to device then the design isn't 
optimized for speed (it's that simple). If I need more speed, in a two stage 
op-amp, I simply decrease the compensation capacitor (to push the unity gain 
frequency, fun = gm/2piCc, out to a higher frequency) and increase the widths 
of the output buffer (keeping the overdrives constant, with an increase in 
current). This pushes f2 (see Eq. [24.24]) to a higher frequency (because 
of the increase in gm2 with the increase in device widths) so that f2 > fun.

Again, for general design set overdrives to 5% of VDD. For high-speed designs
set overdrive voltages to 10% of VDD or more.