Head-related transfer functions (HRTFs) are important for immersive audio, and their spatial interpolation has been studied to upsample finite measurements. Recently, neural fields (NFs) which map from sound source direction to HRTF have gained attention. Existing NF-based methods focused on estimating the magnitude of the HRTF from a given sound source direction, and the magnitude is converted to a finite impulse response (FIR) filter. We propose the neural infinite impulse response filter field (NIIRF) method that instead estimates the coefficients of cascaded IIR filters. IIR filters mimic the modal nature of HRTFs, thus needing fewer coefficients to approximate them well compared to FIR filters. We find that our method can match the performance of existing NF-based methods on multiple datasets, even outperforming them when measurements are sparse. We also explore approaches to personalize the NF to a subject and experimentally find low-rank adaptation to be effective.