An incommensurate elliptical helical magnetic structure in the frustrated coupled-spin-chain system FeTe2O5Br is surprisingly found to persist down to 53(3) mK (T/T-N similar to 1/200), according to neutron scattering and muon spin relaxation. In this state, finite spin fluctuations at T -> 0 are evidenced by muon depolarization, which is in agreement with specific-heat data indicating the presence of both gapless and gapped excitations. We thus show that the amplitude-modulated magnetic order intrinsically accommodates contradictory persistent spin dynamics and long-range order and can serve as a model structure to investigate their coexistence.