Extreme pressures and high magnetic fields can affect materials in profound and fascinating ways. However, large pressures and fields are often mutually incompatible; the rapidly changing fields provided by pulsed magnets induce eddy currents in the metallic components used in conventional pressure cells, causing serious heating, forces, and vibration. Here, we report a diamond-anvil-cell made mainly out of insulating composites that minimizes inductive heating while retaining sufficient strength to apply pressures of up to 8 GPa. Any residual metallic component is made of low-conductivity metals and patterned to reduce eddy currents. The simple design enables rapid sample or pressure changes, desired by pulsed-magnetic-field-facility users. The pressure cell has been used in pulsed magnetic fields of up to 65 T with no noticeable heating at cryogenic temperatures. Several measurement techniques are possible inside the cell at temperatures as low as 500 mK.