I feel strongly there is no way to prove that such thing as a "slowest possible non-zero speed" exists. However there are realistic limits regarding the degree to which slower and slower speeds can reliably be measured.
Example (purely hypothetical)
Consider a pair of neutrinos, ν1 and ν2, emitted from the sun, one right after the other, at times T0 and T1, where:
T1 - T0 = tp (1 Planck time)
Now consider that they travel along the exact same path over the course of 500 billion trillion quadrillion years, and eventually, they arrive at some distant galaxy, where some aliens have a neutrino detector set up. When they get there, they arrive at times T0' and T1', where:
T1' - T0' = 2tp (2 Plank times)
What is the relative velocity difference between the two neutrinos when they were emitted from the sun? It's a really darn small number.
This number could continue getting smaller the farther away the alien detector was, assuming T1' - T0' remained 2tp. Therefore there does not seem to be a minimum possible speed.
Clearly this hypothetical example is ridiculous. I made it that way to illustrate a point, which is that determining a velocity requires two measurements, and it requires being able to identify something as being the same thing you observed both times.
In our example the aliens would have no way to know if the neutrinos both came from exactly the same place or not. Even if they did they'd have no way to observe when the neutrinos were emitted relative to one another.
I have a very strong feeling that the minimum speed that it is possible to observe depends heavily upon the detection equipment, and even then, there are realistic limits to consider.
For example, are we talking about trying to measure the relative speed difference between two objects that are moving at relativistic speeds compared to us? Or are we staring at an apparently immobile object and waiting a million years for it to move a single plank length?
I think there comes a time when an observer would either give up and say, "It ain't moving," or they would die.
Furthermore, instruments used to detect speed are generally calibrated to units of time such as meters per second with a limited degree of accuracy. Something moving 1 plank length per hundred googolplex years would not be considered "in motion," would it?
To put it another way the minimum possible measurable non-zero speed is lp/∂T, where ∂T is the change in time from the first point the object was measured at to the second point the object was measured at, and lp is a planck length.