The epitaxial growth of technically important β-Ga₂O₃ semiconductor thin films has not been realized on flexible substrates due to the limitations of high-temperature crystallization conditions and lattice-matching requirements. We demonstrate the epitaxial growth of β-Ga₂O₃ (-201) thin films on flexible CeO₂ (001)-buffered Hastelloy tape. The results indicate that CeO₂ (001) has a small bi-axial lattice mismatch with β-Ga₂O₃ (-201), inducing simultaneous double-domain epitaxial growth. Flexible photodetectors are fabricated on epitaxial β-Ga₂O₃-coated tape. Measurements reveal that the photodetectors have a responsivity of 4 ×10⁴ mA/W, with an on/off ratio reaching 1,000 under 254 nm incident light and 5 V bias voltage. Such photoelectrical performance is within the mainstream level of β-Ga₂O₃-based photodetectors using conventional rigid single-crystal substrates. More importantly, it remained robust against more than 20,000 bending-test cycles. Moreover, the technique paves the way for the direct in situ epitaxial growth of other flexible oxide semiconductor devices in the future.