On a question of Goss

In this note we answer the question raised by D. Goss in [D. Goss, Applications of non-Archimedean integration to the L-series of τ-sheaves, J. Number Theory 110 (1) (2005) 83–113] by proving that the group of locally analytic endomorphisms on the 1-units of a locally compact field of characteristic p>0 is isomorphic to the p-adic integers.