We introduce a model for a stacked quantum memory made with multi-qubit cells, inspired by multi-level flash cells in classical solid-state drive, and we design quantum error correction codes for this model by generalizing rank-metric codes to the quantum setting. Rank-metric codes are used to correct faulty links in classical communication networks. We propose a quantum generalization of Gabidulin codes, which is one of the most popular family of rank-metric codes, and we design a protocol to correct faults in Clifford circuits applied to a stacked quantum memory based on these codes. We envision potential applications to the optimization of stabilizer states and magic states factories, and to variational quantum algorithms. Further work is needed to make this protocol practical. It requires a hardware platform capable of hosting multi-qubit cells with low crosstalk between cells, a fault-tolerant syndrome extraction circuit for rank-metric codes and an associated efficient decoder.