Reg, G4FGQ wrote:
"What is the voltage measured between the bottom end and ground of a 1
metre high vertical antenna above a perfect ground when the
verticallly-polarized field strength is 1 volt per metre, and antenna
height is shorter than a 1/4-wavelength?"

I see no tricks in the question. The field strength is given as "1 volt
per metre".

From page 23-3 of the 19th edition of the ARRL Antenna Book:
"The standard of measure for field intensities is the voltage developed
in a wire that is 1 meter long, expressed as volts per meter."

While field strength is not the same as the volts delivered to a
receiver, because of the voltage division between antenna impedance and
receiver input impedance, there is no voltage division when the antenna
is loaded with an open circuit. We assume the r-f voltmeter used to
measure voltage at the base of the antenna has an infinite input
impedance. The antenna used for field strength measurements is often a
loop, but we are not concerned with the measurement itself. Reg had a
very simple question, "What`s the voltage at the base of a 1-meter high
wire?

The voltage at the base of an open-circuit 1-metre wire iis one volt
because it goes straight to the definition of field intensity in
volts/m.

Best regards, Richard Harrison, KB5WZI