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Inspired by the success of control barrier functions (CBFs) in addressing safety, and the rise of data-driven techniques for modeling functions, we propose a non-parametric approach for online synthesis of CBFs using Gaussian Processes (GPs). Mathematical constructs such as CBFs have achieved safety by designing a candidate function a priori. However, designing such a candidate function can be challenging. A practical example of such a setting would be to design a CBF in a disaster recovery scenario where safe and navigable regions need to be determined. The decision boundary for safety in such an example is unknown and cannot be designed a priori. In our approach, we work with safety samples or observations to construct the CBF online by assuming a flexible GP prior on these samples, and term our formulation as a Gaussian CBF. GPs have favorable properties, in addition to being non-parametric, such as analytical tractability and robust uncertainty estimation. This allows realizing the posterior components with high safety guarantees by incorporating variance estimation, while also computing associated partial derivatives in closed-form to achieve safe control. Moreover, the synthesized safety function from our approach allows changing the corresponding safe set arbitrarily based on the data, thus allowing non-convex safe sets. We validate our approach experimentally on a quadrotor by demonstrating safe control for fixed but arbitrary safe sets and collision avoidance where the safe set is constructed online. Finally, we juxtapose Gaussian CBFs with regular CBFs in the presence of noisy states to highlight its flexibility and robustness to noise. The experiment video can be seen at: https://youtu.be/HX6uokvCiGk.