In recent years, due to their outstanding dielectric and magnetic properties, hexagonal ferrites (hexaferrites) have attracted considerable interest for developing electronic components of next-generation communication systems. The complex crystal structure of hexaferrites and critical dependences of their electric and magnetic properties on external factors, like magnetic or electric fields, pressure or doping, open ample opportunities for targeted tuning of these properties when designing specific devices. To that end, we explored the electromagnetic properties of the Pb-substituted barium hexaferrite, Ba1-xPbxFe12O19, a compound featuring an extremely rich set of physical phenomena that are inherent in the dielectric and magnetic subsystems of the material and are expected to have significant effect on its electromagnetic response at radio and terahertz frequencies. We performed the first detailed measurements of the AC response of single-crystalline Ba1-xPbxFe12O19 in an extremely broad spectral range from 1 Hz to 240 THz down to temperatures as low as 5 K. We fully characterized numerous microscopic phenomena that determine the broad-band dielectric response of the compound, and we analyzed their nature. This includes temperature-activated radiofrequency relaxations that were attributed to the dynamic response of magnetic/dielectric domains. The terahertz response is dominated by a ferroelectric-like soft mode with an unusual temperature behavior that we explain by means of a microscopic model. Several narrower terahertz excitations are associated with electronic transitions between the fine-structure components of the Fe2+ground state. Narrow resonances detected in the gigahertz region are presumably of magneto-electric origin. The obtained data on diverse but controllable electromagnetic properties of Ba1-xPbxFe12O19 compounds provides the researchers with information that makes the entire class of hexaferrites materials attractive for manufacturing electronic devices for the radiofrequency and terahertz ranges, such as absorbing coatings, anti-reflective coatings, absorbers, electromagnetic shields, antennas, phase shifters, filters, resonators, modulators, etc.