This paper discusses the fundamental combinatorial question of whether or not, for a given string α, there exists a morphism σ such that σ is unambiguous with respect to α, i.e. there exists no other morphism τ satisfying τ(α) = σ(α). While Freydenberger et al. [Int. J. Found. Comput. Sci. 17 (2006) 601–628] characterise those strings for which there exists an unambiguous nonerasing morphism σ, little is known about the unambiguity of erasing morphisms, i.e. morphisms that map symbols onto the empty string. The present paper demonstrates that, in contrast to the main result by Freydenberger et al., the existence of an unambiguous erasing morphism for a given string can essentially depend on the size of the target alphabet of the morphism. In addition to this, those strings for which there exists an erasing morphism over an infinite target alphabet are characterised, complexity issues are discussed and some sufficient conditions for the (non-)existence of unambiguous erasing morphisms are given.