With the aim of representing subsets of Banach spaces as an infinite series using Lipschitz functions, we study a variant of metric frames which we call Lipschitz $p$-approximate Schauder frames (Lipschitz $p$-ASFs). We characterize Lipschitz $p$-ASFs and their duals completely using the canonical Schauder basis for classical sequence spaces. Similarity of Lipschitz $p$-ASF is introduced and characterized.