Long carrier lifetimes and diffusion lengths form the basis for the successful application of the organic-inorganic perovskite (CH$_3$NH$_3$)PbI$_3$ in solar cells and lasers. The mechanism behind the long carrier lifetimes is still not completely understood. Spin-split bands and a resulting indirect band gap have been proposed by theory. Using near band-gap left-handed and right-handed circularly polarized light we induce photocurrents of opposite directions in a single-crystal (CH$_3$NH$_3$)PbI$_3$ device at low temperature ($4~mathrm{K}$). The phenomenom is known as the circular photogalvanic effect and gives direct evidence for phototransport in spin-split bands. Simultaneous photoluminecence measurements show that the onset of the photocurrent is below the optical band gap. The results prove that an indirect band gap exists in (CH$_3$NH$_3$)PbI$_3$ with broken inversion symmetry as a result of spin-splittings in the band structure. This information is essential for understanding the photophysical properties of organic-inorganic perovskites and finding lead-free alternatives. Furthermore, the optically driven spin currents in (CH$_3$NH$_3$)PbI$_3$ make it a candidate material for spintronics applications.
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