Two pairs of positive-and negative-parity doublet bands together with eight strong electric dipole transitions linking their yrast positive- and negative-parity bands have been identified in ^{78}Br. They are interpreted as multiple chiral doublet bands with octupole correlations, which is supported by the microscopic multidimensionally-constrained covariant density functional theory and triaxial particle rotor model calculations. This observation reports the first example of chiral geometry in octupole soft nuclei.