The $\beta$ decay of the odd-odd nucleus $^{70}$Br has been investigated with the BigRIPS and EURICA setups at the Radioactive Ion Beam Factory (RIBF) of the RIKEN Nishina Center. The $T=0$ ($J^{\pi}=9^+$) and $T=1$ ($J^{\pi}=0^+$) isomers have both been produced in in-flight fragmentation of $^{78}$Kr with ratios of 41.6(8)\% and 58.4(8)\%, respectively. A half-life of $t_{1/2}=2157^{+53}_{-49}$ ms has been measured for the $J^{\pi}=9^+$ isomer from $\gamma$-ray time decay analysis. Based on this result, we provide a new value of the half-life for the $J^{\pi}=0^+$ ground state of $^{70}$Br, $t_{1/2}=78.42\pm0.51$ ms, which is slightly more precise, and in excellent agreement, with the best measurement reported hitherto in the literature. For this decay, we provide the first estimate of the total branching fraction decaying through the $2^+_1$ state in the daughter nucleus $^{70}$Se, $R(2^+_1)=1.3\pm1.1\%$. We also report four new low-intensity $\gamma$-ray transitions at 661, 1103, 1561, and 1749 keV following the $\beta$ decay of the $J^{\pi}=9^+$ isomer. Based on their coincidence relationships, we tentatively propose two new excited states at 3945 and 4752 keV in $^{70}$Se with most probable spins and parities of $J^{\pi}=(6^+)$ and $(8^+)$, respectively. The observed structure is interpreted with the help of shell-model calculations, which predict a complex interplay between oblate and prolate configurations at low excitation energies.Read less <