Non-binary codes correcting multiple deletions have recently attracted a lot of attention. In this work, we focus on multiplicity-free codes, a family of non-binary codes where all symbols are distinct. Our main contribution is a new explicit construction of such codes, based on set and permutation codes. We show that our multiplicity-free codes can correct multiple deletions and provide a decoding algorithm. We also show that, for a certain regime of parameters, our constructed codes have size larger than all the previously known non-binary codes correcting multiple deletions.