Because, hypothetically, it could act like a capacitor, or charge storage device, and the electric field in the tube may take time to discharge?
Because, hypothetically, the electrons surrounding the sodium atoms that have been raised to a higher potential by the electric field may take time to decay back to the ground state (there may be a time constant for that decay).
I do not know the physics, and the above hypotheses may be wrong, but I do not see a basis for simply ignoring the possibility without evidence to the contrary.
To indulge:
1. Nah, no charge, it's a heated ceramic tube with ionized stuff inside, so the resistance will stay fairly low during a cycle (compared to cold resistance that is). So, microseconds there -- limited more by the driver transformer inductance or capacitance, say.
2. Deionization time is definitely a thing. It's not a charge separation thing (the plasma is very nearly neutral at all times and in all places, except for thin boundary regions near the electrodes, if applicable). It's more of a thermal mass thing. Plasma is the "4th state of matter", though it doesn't have a sharp phase change as the other states do (it's not a first order transition, maybe not even a 2nd order transition, I'm not sure). Basically, it takes time for it to cool down, that is, for the ionized atoms to recombine and give off whatever energy they do (which for sodium lamps, happens to be a very large fraction of emitted 589nm light!).
There are various factors for DI time, and I don't know much about it. Heavy ions and low densities tend to take longer. Example, low pressure xenon and mercury thyratrons are typically ~1ms. (Presumably, you can see the glow discharge dissipate over the same time scale. Hmm, I should give that a try.) Neon lamps, more like 100us. Hydrogen thyratrons, us to ns.
Another way you can experience recombination time: sodium in a flame. A flame is hydrocarbon radicals and various intermediate combustion products, all emitting at various wavelengths (with the CH. and related species being the most notable, in the blue-green range, for low C:H ratio (gas fuel), neutral flames. Well, clearly those reactions are delivering a good bit of energy, if they're sometimes giving off blue light! Introduce some sodium atoms and they'll pick up that energy no problem. An ionic flame is also conductive, and, yes, it can even be used as a triode! Well, both the conductivity and light emission only last as long as the atoms have enough energy to do so, so you see a streak of bright orange glow in the flame, until it cools (or finishes reacting) enough to give essentially no more emission. You can calculate the flame velocity and see how long the sodium is (partially) ionized.
Though this will probably give an upper limit (a relatively large value like 10s of ms), because all the other things are reacting much slower than the sodium atoms themselves.
So, that's that.
One could settle this very easily and certainly by watching the spectrum of the lamp under these conditions -- dollars to donuts, it goes from sodium d-line to black body (orange hot) when switched off.
Another point in that favor: the afterglow looks damn like a thermal time constant (seconds, while nothing could possibly be storing that much electrical energy).
Oh, hey, so, on a related subject, long-life phosphors are kind of like very slow recombination. Similar physics (well, hand-wavingly so), but in solid state instead, of course.
There are actually a few gasses with very long transition times, like singlet oxygen (which emits at deep red / near IR), around 30 seconds half life when prepared pure, IIRC. This is possible due to spin transition rules prohibiting the singlet-triplet transition in molecular oxygen (triplet ground state), the lack of impurities with which to exchange those spins, and the slow reaction rate on container walls (because surface area is small compared to the volume). None of these are relevant to the contents of a sodium bulb, but if you don't know that, you're actually not at all wrong to suspect that it might be possible, because, in fact, it is; just that it's limited to much more contrived conditions and specific molecules.
Tim