Scalability remains one of the biggest challenges to the adoption of permissioned blockchain technologies for large-scale deployments. Permissioned blockchains typically exhibit low latencies, compared to permissionless deployments -- however at the cost of poor scalability. Various solutions were proposed to capture "the best of both worlds", targeting low latency and high scalability simultaneously, the most prominent technique being blockchain sharding. However, most existing sharding proposals exploit features of the permissionless model and are therefore restricted to cryptocurrency applications. We present MITOSIS, a novel approach to practically improve scalability of permissioned blockchains. Our system allows the dynamic creation of blockchains, as more participants join the system, to meet practical scalability requirements. Crucially, it enables the division of an existing blockchain (and its participants) into two -- reminiscent of mitosis, the biological process of cell division. MITOSIS inherits the low latency of permissioned blockchains while preserving high throughput via parallel processing. Newly created chains in our system are fully autonomous, can choose their own consensus protocol, and yet they can interact with each other to share information and assets -- meeting high levels of interoperability. We analyse the security of MITOSIS and evaluate experimentally the performance of our solution when instantiated over Hyperledger Fabric. Our results show that MITOSIS can be ported with little modifications and manageable overhead to existing permissioned blockchains, such as Hyperledger Fabric.