The solid-state pulsed power generators (PPGs) are widely used in various modern industrial applications for their high repetition rate and long lifetime. The output characteristics of the generators are significantly affected by the dynamic characteristics of the semiconductor switches. Wide bandgap (WBG) semiconductor power devices, such as silicon carbide metal-oxide-semiconductor field-effect transistors (SiC MOSFETs), have the potential to structure solid-state PPGs due to many advantages, such as a high rated voltage, low on-resistance, high operational temperature, and fast switching speed. Typically, these devices are used in series/parallel connections to enhance the power level of the generators. However, this approach often leads to increased parasitic inductance of the switching modules, which results in flattened rising edges of the output pulse and reduced power transmission efficiency. The chip-on-board (COB) packaging method exhibits excellent high-frequency performance by allowing current paths to be more flexible, which is suitable for constructing the power module. This article proposes a solid-state PPG with Blumlein pulse forming network (BPFN) and COB packaging SiC-based switching module. The switching module is based on a four-layer printed circuit board (PCB). SiC MOSFET bare dies, gate drivers, and auxiliary elements are directly soldered on the top layer of the PCB. Separate conducting layers are connected to the terminals of the switching module by vias, which provides extra flexibility in designing and enables the stray inductance of the power loop to be minimized to 5.14 nH by optimizing the current communication loops. A 5-kV prototype of the solid-state PPG is fabricated based on the designed switching module. Under a resistive load of 2.8 $\Omega$ , a quasi-square pulse with a 62-ns pulsewidth and a 45-ns rise time can be obtained. Additionally, this generator has been tested at a 500-Hz repetition rate in the burst mode, and the di/dt of the switching module is about 80 A/ns, proving the effectiveness of the solid-state PPG.