The (Model-Free) Data-Driven Computational Mechanics (DDCM) \cite{Ortiz} approach differs from classical PDE-based methods by replacing constitutive models with raw material data, leading to a variational formulation based on a double minimisation between mechanically admissible states and data-consistent states. While several admissibility constraints can be handled naturally within a finite element framework, others require dedicated treatment in the DDCM setting. In this work, we address unilateral constraints arising in contact mechanics, expressed as nonlinear Signorini-type complementarity conditions. Despite their practical importance and nontrivial numerical nature, such constraints have not yet been considered in the specialised literature, to the best of the authors’ knowledge. The imposition of the nonlinear complementary constraints is treated using the Fischer-Burmeister function \cite{Fischer}, as an alternative to classical methods such as penalisation or augmented Lagrangian approaches, which are less straightforward to implement in a DDCM-based formulation. After introducing a one-dimensional toy problem to demonstrate the core ideas, we present a three-dimensional finite-strain problem for frictionless contact, implemented in ddfenics (to be presented in MS031 on Open-Source Software in Mechanics), to showcase the full capabilities of the method. The proposed formulation extends DDCM to contact problems and paves the way for data-driven simulations of systems with other types of inequality constraints. Future applications include contact mechanisms at both micro- and macro-scales, in particular through recent formulations that allow computational homogenisation to be accelerated by leveraging DDCM.