The strong Whitney convergence on bornology introduced by Caserta in [9] is a generalization of the strong uniform convergence on bornology introduced by Beer-Levi in [5]. This paper aims to study some important topological properties of the space of all real valued continuous functions on a metric space endowed with the topologies of Whitney and strong Whitney convergence on bornology. More precisely, we investigate metrizability, various countability properties, countable tightness, and Fréchet property of these spaces. In the process, we also present a new characterization for a bornology to be shielded from closed sets.