- We propose a pulsar radio emission mechanism that involves a plasma instability not previously considered in the context of pulsars: a non-resonant, beam-driven, hydrodynamical instability in a one-dimensional, highly relativistic, streaming, pair plasma. The growing waves are in a beam mode at frequencies below the frequency of the (known) analogous resonant instability. The instability is analysed in detail for a cold plasma and a cold beam, and the inclusion of a random relativistic spread in momenta does not change the conclusions substantially. The net amplification caused by the non-resonant instability is much larger than for the resonant instability owing to its broad-band nature, implying growth over a much greater distance, as the ratio of the wave frequency to the resonant frequency decreases through the inhomogeneous pulsar magnetosphere. When this ratio reaches unity the beam mode joins on to the L-O mode, and the waves subsequently freely escape from the magnetosphere. Similar to other beam instabilities, effective growth requires a sufficiently dense beam of not too high-energy particles.