The mixed anion copper compound Cu(OH)F was studied in measurements of dc- and ac-magnetic susceptibility, static and pulsed field magnetization, specific heat, X-band electron magnetic resonance and muon-spin spectroscopy. In variance with its layered structure, the magnetic behavior shows no evidence of low-dimensionality. Cu(OH)F reaches short range static antiferromagnetic order at T_N = 9.5-11.5 K and experiences the spin-flop transition at B ∼ 3.5 T. This behavior is in a sharp contrast with physical properties of earlier reported isostructural compound Cu(OH)Cl. The first principle calculations reveal highly competitive nature of ferromagnetic and antiferromagnetic superexchange interactions, the details being rather sensitive to choice of magnetic structure employed in the extraction of magnetic interaction. Rather broad anomaly in C_p(T) dependence at phase transition and smeared magnetization curve M(B) at low temperatures suggest static disorder at low temperatures however no frequency dependence of the magnetization peak inherent for a spin-glass state was detected. Zero-field μSR data confirm transition into a static magnetically disordered state. The inter-substitutions of (OH)^- and F^- ions and relevant variations of exchange interaction parameters may be responsible for the formation of this peculiar ground state.