We report the microwave response of coplanar-waveguide (CPW) resonators fabricated from YBa2Cu3O7−δ (YBCO) thin films over temperatures from approximately 70 mK to 40 K. The resonators exhibit internal quality factors Qi in the range of 4 × 10^3 to 10^4 at 70 mK, which increase to a maximum of approximately 8 × 10^3 to 1.2 × 10^4 near 6 K. At low temperatures, both Qi and the fractional shift of the resonance frequency ∆fr/fr increases with temperature, qualitatively resembling behavior commonly associated with two-level-system (TLS) defects. However, neither response saturates on the temperature scale set by the resonator frequency, and the loss exhibits no observable microwave-power dependence. We show that low-temperature frequency upturn may be better described by an additional paramagnetic response associated with defect-induced local moments or Andreev bound states, while the low-temperature loss follows an approximately logarithmic temperature dependence whose microscopic origin remains unresolved. These measurements establish the millikelvin performance of patterned YBCO resonators and show that their low-temperature response cannot be understood within the conventional TLS framework alone.