  Call a colouring of a graph \emph{distinguishing} if the only automorphism of
this graph which preserves said colouring is the identity. Let $H$ be an
arbitrary graph. We say that a graph $G$ is \emph{$H$-free} if $G$ does not
contain an induced subgraph isomorphic to $H$. Kargul, Musia{\l}, Pal and
Gorzkowska showed that if $n$ is a natural number greater than two, then every
finite connected $K_{1,n}$-free graph of order at least six admits a
distinguishing edge colouring with at most $n-1$ colours. We extend this result
to all locally finite connected $K_{1,n}$-free graphs of order at least six.
