Thank you for your replies.
The wiki article was of course my first recourse: I simply didn't understand the maths, and for the rest, could not see how the information there had a bearing on my specific question. (I've got a lot of respect for Wikipedia's technical material, actually, and I've often found it to be more accurate than printed sources, but also tend to find that it only makes sense if you already know what it is saying).
There must be some current available when you press the disk, no? Just a very small amount. I mean, as long as there is a voltage, there must be a charge, and if there's a charge, there must be the possibility of a current.
During the time I am pressing the disk down, I am continuing to expend energy (sometimes I increase the force, to see if I can stop the voltage from dropping again). So I don't see why the voltage should not stay at the high level. It's not that I'm expecting infinite energy: I expect it to stop sometime, after I let go.
OK, so if it's my probe that's causing the minuscule charge to drain off, I tried testing that: I didn't touch the disk with my probe until a few seconds after I had started pressing the disk. The voltage reading was near zero, so either the charge had leaked off some other way (e.g. into the breadboard) or the theory was wrong.
Concerning my final question (what happens when I make a circuit by connecting a capacitor to the terminals of the disk?), I should add that I chose 47nF for the capacitor because it was just over twice the capacitance I measured from the disk itself, with my multimeter: 18nF. My mind is still unsettled by the question. If the voltage rises on the piezo, some charge must flow into the capacitor, so the cap voltage will also rise; but as the charge flows out of the piezo, the piezo voltage must fall. I guess they'll meet in the middle, and the capacitor will always lag slightly behind (but there's no resistor in the circuit, so it must appear almost instantaneous): and the difference in capacitances might explain the 'slowing down' effect I observed? Also does the relative size of the two capacitances impose a ceiling on the voltage produced by the piezo?
Please note, my questions are very specifically about continued pressure on a disk. I have spent three days trying to get general information from 'introductions' and 'how-it-works' type sources. Eventually you just have to ask someone, don't you?