Superconductivity was recently observed in iron-arsenic-based compounds with a superconducting transition temperature (T _c) as high as 56 K, naturally raising comparisons with the high-T _c copper oxides. The copper oxides have layered crystal structures with quasi-two-dimensional electronic properties, which led to speculation that reduced dimensionality (that is, extreme anisotropy) is a necessary prerequisite for superconductivity at temperatures above 40 K. Early work on the iron-arsenic compounds seemed to support this view. Here we report measurements of the electrical resistivity in single crystals of (Ba,K)Fe₂As₂ in a magnetic field up to 60 T. We find that the superconducting properties are in fact quite isotropic, being rather independent of the direction of the applied magnetic fields at low temperature. Such behaviour is strikingly different from all previously known layered superconductors, and indicates that reduced dimensionality in these compounds is not a prerequisite for 'high-temperature' superconductivity. We suggest that this situation arises because of the underlying electronic structure of the iron-arsenic compounds, which appears to be much more three dimensional than that of the copper oxides. Extrapolations of low-field single-crystal data incorrectly suggest a high anisotropy and a greatly exaggerated zero-temperature upper critical field.