This paper presents a polar-format algorithm (PFA)-based synthetic-aperture radar (SAR) processor that can be mounted on a small drone to support video SAR (ViSAR) imaging. For drone mounting, it requires miniaturization, low power consumption, and high-speed performance. Therefore, to meet these requirements, the processor design was based on a field-programmable gate array (FPGA), and the implementation results are presented. The proposed PFA-based SAR processor consists of both an interpolation unit and a fast Fourier transform (FFT) unit. The interpolation unit uses linear interpolation for high speed while occupying a small space. In addition, the memory transfer is minimized through optimized operations using SAR system parameters. The FFT unit uses a base-4 systolic array architecture, chosen from among various fast parallel structures, to maximize the processing speed. Each unit is designed as a reusable block (IP core) to support reconfigurability and is interconnected using the advanced extensible interface (AXI) bus. The proposed PFA-based SAR processor was designed using Verilog-HDL and implemented on a Xilinx UltraScale+ MPSoC FPGA platform. It generates an image 2048 × 2048 pixels in size within 0.766 s, which is 44.862 times faster than that achieved by the ARM Cortex-A53 microprocessor. The speed-to-area ratio normalized by the number of resources shows that it achieves a higher speed at lower power consumption than previous studies.