Global mean sea level rise and its acceleration are projected to aggravate coastal erosion over the 21st century, which constitutes a major challenge for coastal adaptation. Projections of shoreline retreat are highly uncertain, however, namely due to deeply uncertain mean sea level projections and the absence of consensus on a coastal impact model. An improved understanding and a better quantification of these sources of deep uncertainty are hence required to improve coastal risk management and inform adaptation decisions. In this work we present and apply a new extra-probabilistic framework to develop shoreline change projections of sandy coasts that allows consideration of intrinsic (or aleatory) and knowledge-based (or epistemic) uncertainties exhaustively and transparently. This framework builds upon an empirical shoreline change model to which we ascribe possibility functions to represent deeply uncertain variables. The model is applied to two local sites in Aquitaine (France) and Castellón (Spain). First, we validate the framework against historical shoreline observations and then develop shoreline change projections that account for possible (although unlikely) low-end and high-end mean sea level scenarios. Our high-end projections show for instance that shoreline retreats of up to 200 m in Aquitaine and 130 m in Castellón are plausible by 2100, while low-end projections revealed that 58 and 37 m modest shoreline retreats, respectively, are also plausible. Such extended intervals of possible future shoreline changes reflect an ambiguity in the probabilistic description of shoreline change projections, which could be substantially reduced by better constraining sea level rise (SLR) projections and improving coastal impact models. We found for instance that if mean sea level by 2100 does not exceed 1 m, the ambiguity can be reduced by more than 50 %. This could be achieved through an ambitious climate mitigation policy and improved knowledge on ice sheets.