FWIW when you're closer to an antenna than the wavelength, you're not just seeing normal Gauss's law drop off, you can be looking an an evanescent wave [1], which does indeed fall off exponentially (well, depends on the exact shape of the antenna).

I've designed a short-range 'antenna' that is intentionally not impedance matched to the vacuum, and radiates very poorly - but the RF intensity close to the antenna is very high because of this effect.

[1] https://en.m.wikipedia.org/wiki/Evanescent_field