The iron- and nickel-based layered compounds LaOFeP (refs 1, 2) and LaONiP (ref. 3) have recently been reported to exhibit low-temperature superconducting phases with transition temperatures T(c) of 3 and 5 K, respectively. Furthermore, a large increase in the midpoint T(c) of up to approximately 26 K has been realized in the isocrystalline compound LaOFeAs on doping of fluoride ions at the O2- sites (LaO(1-x)F(x)FeAs). Experimental observations and theoretical studies suggest that these transitions are related to a magnetic instability, as is the case for most superconductors based on transition metals. In the copper-based high-temperature superconductors, as well as in LaOFeAs, an increase in T(c) is often observed as a result of carrier doping in the two-dimensional electronic structure through ion substitution in the surrounding insulating layers, suggesting that the application of external pressure should further increase T(c) by enhancing charge transfer between the insulating and conducting layers. The effects of pressure on these iron oxypnictide superconductors may be more prominent than those in the copper-based systems, because the As ion has a greater electronic polarizability, owing to the covalency of the Fe-As chemical bond, and, thus, is more compressible than the divalent O2- ion. Here we report that increasing the pressure causes a steep increase in the onset T(c) of F-doped LaOFeAs, to a maximum of approximately 43 K at approximately 4 GPa. With the exception of the copper-based high-T(c) superconductors, this is the highest T(c) reported to date. The present result, together with the great freedom available in selecting the constituents of isocrystalline materials with the general formula LnOTMPn (Ln, Y or rare-earth metal; TM, transition metal; Pn, group-V, 'pnicogen', element), indicates that the layered iron oxypnictides are promising as a new material platform for further exploration of high-temperature superconductivity.