Antennas do have voltages induced across them by
being in a space exposed to a passing electromagnetic
wave. The result is currents in the antenna elements,
and hence we tap in to recover some energy to be
the input for our receivers.
Operating in
transmit mode, we can deliver energy to antennas,
and excite currents in the elements, and corresponding
voltages across them. The resulting electric and
magnetic fields around the antenna are local to
it (near field), but in the space up to 2 wavelengths
away, they do generate the radio wave electromagnetic
field which travels from the antenna at the speed
of light.
I once fed a modest power of about
100W to a whip antenna which was a conductor encased
in fibreglass. The voltage on the end of the antenna
was high enough to cause air corona breakdown at
the tip, and the resin binder caught fire. If you
need convincing that antennas have voltages on them,
try holding up a old flourescent tube near the end
of a ham HF antenna while he is transmitting, and
see it light up!