Metastatic tumours often invade healthy neighbouring tissues by forming multicellular finger-like protrusions emerging from the cancer mass. To understand the mechanical context behind this phenomenon, we here develop a minimalist fluid model of a self-propelled, growing biological tissue. The theory involves only four mechanical parameters and remains analytically trackable in various settings. As an application of the model, we study the evolution of a two-dimensional circular droplet made of our active and expanding fluid, and embedded in a passive non-growing tissue. This system could be used to model the evolution of a carcinoma in an epithelial layer. We find that our description can explain the propensity of tumour tissues to fingering instabilities, as conditioned by the magnitude of active traction and the growth kinetics. We are also able to derive predictions for the tumour size at the onset of metastasis, and for the number of subsequent invasive fingers. Our active fluid model may help describe a wider range of biological processes, including wound healing and developmental patterning.