We say that a topological monoid $S$ is left non-archimedean (in short: l-NA) if the left action of $S$ on itself admits a proper $S$-compactification $\nu \colon S \hookrightarrow Y$ such that $Y$ is a Stone space. This provides a natural generalization of the well known concept of NA topological groups. In this note we give several results which naturally extend some known results regarding NA topological groups. The Stone and Pontryagin dualities play major role in achieving useful characterizations of NA monoids. We discuss universal NA monoids and show that many naturally defined topological monoids are NA.