Previous efforts on making Satisfiability (SAT) solving fit for high performance computing (HPC) have led to super-linear speedups on particular formulae, but for most inputs cannot make efficient use of a large number of processors. Moreover, long latencies (minutes to days) of job scheduling make large-scale SAT solving on demand impractical for most applications. We address both issues with Mallob, a framework for job scheduling in the context of SAT solving which exploits malleability, i.e., the ability to add or remove processing power from a job during its computation. Mallob includes a massively parallel, distributed, and malleable SAT solving engine based on HordeSat with a more succinct and communication-efficient approach to clause sharing and numerous further improvements over its precursor. For example, Mallob on 640 cores outperforms an updated and improved configuration of HordeSat on 2560 cores. Moreover, Mallob can solve many formulae in parallel while dynamically adapting the assigned resources, and jobs arriving in the system are usually initiated within a fraction of a second.