In this letter, we report on the realization of a two-stage 16-way solid-state power amplifier (SSPA) in the Ka-band. To this end, we describe the design of a high-power amplifier (HPA) in a 100-nm gallium nitride (GaN) process and its integration into a split-block waveguide module. The PA module achieves an output power of more than 7.6 W between 28 and 39 GHz. In conjunction with 16 of these PA modules, we then employed a custom low-loss radial splitter and combiner to create a compact SSPA system. The two-stage SSPA configuration exhibits a small-signal gain of up to 44 dB and a peak output power of 127 W at 31 GHz in 5 dB of gain compression. Furthermore, we measured output power of close to 100 W and state-of-the-art efficiency values of more than 19% between 28 and 38 GHz. To our knowledge, this is the most broadband high-power SSPA demonstrated so far in this frequency range.