Do particles have spatial extent or are they point-like?

It seems to me that if this were true, then FTL communication would be possible.

As a thought-experiment, consider an experimenter who has a container PACKED full of photons. He can release photons at will so that they collide with a nuclei, and via pair production, this produces an electron and a positron.

Let's say we want to transmit the message "101" faster than the speed of light to an observer 100 meters away, who has an electron detector.

At t=0, the experimenter releases a huge number of these photons all at once. By pair production, an equally huge number of electrons is produced. At the instant these electrons are produced, the observer 100 meters away would INSTANTLY register a statistical anomaly at his electron detector, simply because as the number of electrons produced approaches infinity, the probability that the observer detects one of these at his detector approaches 1.

At t=1, the experimenter does nothing. The observer notices no statistically significant change in his data at this time.

At t=2, the experimenter again releases a huge number of photons. The observer now would see another statistical anomaly.

So, how do we explain this then? The only thing I can think of is that the probability domain of measuring an electron is not the entire universe and that the theory is either too simplified to make it easy to understand or that there is something fundamentally wrong with it (I'm guessing the former).

"the number of electrons produced approaches infinity"
It doesn't approach infinity at all.

"the probability that the observer detects one of these at his detector approaches 1"
Only if they had enough time to travel those 100 meters, with a speed less than c, else he won't detect any of the electrons.

A particle can be anywhere, as long as it is possible for it to get there, given the observations that were previously made on it.