Discrete Global Grid Systems (DGGS) provide a hierarchical alternative to conventional raster and vector spatial representations, offering globally consistent indexing, equal-area tessellations, and native support for multi-resolution analysis. Although DGGS have been widely applied in Earth observation and geospatial analytics, comparatively little work has explored their use as the primary computational mesh for hydrodynamic flood simulation. This paper presents FloodA5, an open-source flood modelling framework built on the A5 equal-area pentagonal DGGS. FloodA5 integrates DGGS-native mesh generation, terrain processing, hydrodynamic simulation, sub-grid terrain representation, data storage, and visualisation within a unified workflow implemented in Julia and Python. The framework employs an inertial formulation of the shallow-water equations adapted for operation on a non-orthogonal pentagonal mesh and includes an optional sub-grid sampling (SGS) approach based on pre-computed hypsometric storage relationships. FloodA5 is evaluated using synthetic benchmark problems and the Carlisle 2005 flood event. Results demonstrate that physically plausible flood simulations can be produced on an equal-area pentagonal DGGS and show reasonable agreement with a LISFLOOD-FP reference simulation, with the best-performing configuration achieving an intersection-over-union score of 0.67 and a depth RMSE of 1.17 m. Synthetic benchmarks reveal directional routing artefacts associated with the current treatment of non-orthogonal gradients and limitations in representing internal topographic barriers within the SGS formulation. The results demonstrate the feasibility of DGGS-native flood modelling while highlighting the importance of numerical schemes specifically designed for non-orthogonal polygonal meshes. FloodA5 provides an open-source platform for future research into hydrodynamic simulation on hierarchical global grid systems.