The hydrolysis of borohydride salts represents one of the most promising processes for the generation of high purity molecular hydrogen under mild conditions. In this work we show that the sodium borohydride hydrolysis exhibits a fingerprinting periodic oscillatory transient in the hydrogen flow over a wide range of experimental conditions. We disproved the possibility that flow oscillations are driven by supersaturation phenomena of gaseous bubbles in the reactive mixture or by a nonlinear thermal feedback according to a thermokinetic model. Our experimental results indicate that the NaBH4 hydrolysis is a spontaneous inorganic oscillator, in which the hydrogen flow oscillations are coupled to an "oscillophor" in the reactive solution. The discovery of this original oscillator paves the way for a new class of chemical oscillators, with fundamental implications not only for testing the general theory on oscillations, but also with a view to chemical control of borohydride systems used as a source of hydrogen based green fuel.