Zipper fractal interpolation function (ZFIF) is a generalization of fractal interpolation function through an improved version of iterated function system by using a binary parameter called a signature. The signature allows the horizontal scalings to be negative. ZFIFs have a complex geometric structure, and they can be non-differentiable on a dense subset of an interval $I$. In this paper, we construct $k$-times continuously differentiable ZFIFs with variable scaling functions on $I$. Some properties like the positivity, monotonicity, and convexity of a zipper fractal function and the one-sided approximation for a continuous function by a zipper fractal function are studied. The existence of Schauder basis of zipper fractal functions for the space of $k$-times continuously differentiable functions and the space of $p$-integrable functions for $p \in [1,\infty)$ are studied. We introduce the zipper versions of full Müntz theorem for continuous function and $p$-integrable functions on $I$ for $p \in [1,\infty)$.