Differential Dynamic Programming (DDP) is a powerful optimisation method for space trajectory design, known for its speed, robustness to poor initialisation, and versatility in addressing a wide range of problems marked by highly nonlinear dynamics, as encountered in astrodynamics. The approach leverages Bellman's principle of optimality and quadratic approximations of the cost function for iterative refinement of design variables at each iteration. Incorporating Taylor Differential Algebra (DA) techniques can significantly enhance DDP runtimes. These techniques entail mapping functions over a continuous set using Taylor polynomials, enabling efficient evaluation of the mapped function's value within a given convergence radius through inexpensive polynomial evaluations. Additionally, the derivatives of the functions, assuming sufficient differentiability, can be obtained with limited effort. Building on these principles, this study introduces DA-based enhancements to DDP. The resulting DA-based DDP (DADDy) solver enforces nonlinear constraints using an Augmented Lagrangian formulation. Upon finding a preliminary solution, it undergoes "polishing" via a projected Newton method to verify constraints up to machine precision. Validation of the solver is conducted with a fuel-optimal low-thrust Earth-Mars transfer. Due to inherent nonlinearities, an initial computation yields a more regular minimum-energy solution, forming the basis for iteratively determining the minimum-fuel solution. Results demonstrate that the DADDy solver achieves comparable outcomes to classic DDP, with runtimes reduced by 61% to 93%, depending on the dynamics.