Modern GPUs equipped with mixed precision tensor core units present great potential to accelerate dense linear algebra operations such as LU factorization. However, previous works have focused solely on improving speed, neglecting memory consumption. Indeed, state-of-the-art mixed half/single precision LU factorization algorithms all require the matrix to be stored in single precision. This is explained by the fact that simply switching the storage precision from single to half leads to significant loss of accuracy, forfeiting all accuracy benefits from using tensor core technology. In this article, we propose a new factorization algorithm that is able to store the matrix in half precision without incurring any significant loss of accuracy. Our approach is based on a left-looking scheme employing single precision buffers of controlled size and a mixed precision doubly partitioned algorithm exploiting tensor cores in the panel factorizations. Our numerical results show that compared with the state of the art, the proposed approach is of similar accuracy, up to twice faster, and with only half the data movement and memory footprint.