In this work, we present an on-chip Hafnium Zirconium Oxide-based millimeter wave (mmWave) programmable antenna. The device exhibits a resonance frequency of 36.623 GHz, which can be programmed by varying the polarization state of a thin Hafnium Zirconium Oxide (HZO) layer enclosed in the structure of the antenna. By polarizing such HZO film, we demonstrate, for the first time, a variation in the resonance frequency for ferroelectrically programmable antennas without continuously applying a bias. In this regard, the non-volatile behavior of HZO allows to retain the polarization state of the film even after removing the DC voltage, therefore enabling the antenna's resonance frequency retention. The full ferroelectric switching of the HZO layer occurs approximately at ±3 V, resulting in a maximum resonance frequency shift of 3.019 GHz (8.3% fractional frequency change) between different programming voltages. The device's response to the applied programming voltage generates a ferroelectrically-induced hysteresis on its resonance frequency. In contrast to prior devices of this class, the proposed approach showcases a more compact size, full post-CMOS compatibility, low programming voltage, and non-volatile programmability.