Swelling of individual fibres exposed to a favorable solvent may affect the mechanical properties or shape of a fibrous material. We provide experimental results of the absorption dynamics of a single drop deposited on a swellable fibre and show that the total absorption time is highly dependent on the fibre geometry and drop volume. The curvature of the fibre prevents the total spreading of a drop even if the fluid is fully wetting the fibre. For drops larger than a critical volume, a local saturation of the fibre is thus reached below the drop, leading to a strong increase in the total absorption time. These observations are then rationalized with a simple pseudo-diffusive model to understand the drop absorption and fibre swelling dynamics. This minimal model provides quantitative predictions of the total absorption time.