The heterogeneous micromechanical properties of biological tissues have profound implications across diverse medical and engineering domains. However, identifying the full-field heterogeneous elastic properties of soft materials using traditional computational and engineering approaches is fundamentally challenging due to difficulties in estimating local stress fields. Recently, there has been a growing interest in using data-driven models to learn full-field mechanical responses such as displacement and strain from experimental or synthetic data. However, research studies on inferring the full-field elastic properties of materials, a more challenging problem, are scarce, particularly for large deformation, hyperelastic materials. Here, we propose a physics-informed machine learning approach to identify the elastic modulus distribution in nonlinear, large deformation hyperelastic materials. We evaluate the prediction accuracies and computational efficiency of physics-informed neural networks (PINNs) on inferring the heterogeneous material parameter maps across three nonlinear materials with structural complexity that closely resemble real tissue patterns, such as brain tissue and tricuspid valve tissue. Our improved PINN architecture accurately estimates the full-field elastic properties of three hyperelastic constitutive models, with relative errors of less than 5% across all examples. This research has significant potential for advancing our understanding of micromechanical behaviors in biological materials, impacting future innovations in engineering and medicine.