Locally-APN functions are a generalization of APN power functions which satisfy the APN condition only locally. In this paper, we naturally extend the original definition of locally-APN functions from power functions to arbitrary functions on \(\mathbb {F}_2^n\) . We then present a new construction of locally-APN functions which uses a spread and a Sidon set. This combinatorial approach leads to a large family of locally-APN functions containing several known locally-APN power functions. We precisely determine the differential spectrum and the algebraic degree of the functions in this family.