On a question of Goss

In this note we answer the question raised by D. Goss in [Applications of non-Archimedean integration to the $L$-series of $\tau$-sheaves, {\em J. Number Theory,} 110 (2005), no. 1, 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.