The first generation of S-band injectors for free-electron lasers has illustrated high reliability and performance by driving the latest generation of high brightness machines. However, the ultimate electron beam brightness of these devices is limited by the electric field that they can achieve on the cathode. With the aim of a higher beam brightness at SwissFEL, this paper presents the design of a C-band traveling-wave (TW) radio-frequency photogun that aims to offer the possibility to move to a cathode gradient up to 200  MV/m. This increased gradient comes from the ability to operate with rf pulse lengths approximately an order of magnitude shorter than the current state-of-the-art room-temperature S-band standing-wave rf photoguns. With this high cathode gradient, this novel gun is able to produce an electron bunch whose 5D beam brightness is 5 times greater than the current SwissFEL photogun. Furthermore, the low power dissipation within the photogun, resulting from the short rf pulse length and TW philosophy, opens up the opportunity for rf pulse repetition rates up to 1 kHz.