Let $M$ be a Stein manifold with smooth, strictly
pseudoconvex boundary, and $x$ a point on its
boundary. Is there a holomorphic function $f$ on
$M$, smooth on the boundary, with strict
maximum of $|f|$ in $x$?

This question was treated for domains in ${\mathbb C}^n$,
but I could not find any reference for bigger generality.

<b>Update</b>. The set of maxima is called "Shilov boundary".
For domains in ${\mathbb C}^n$, Shilov boundary is the
set of strictly pseudoconvex points on the boundary, as follows
from the papers 

Richard F. Basener,
<i>Peak Points, Barriers and Pseudoconvex Boundary Points,</i>
Proceedings of the American Mathematical Society Vol. 65, No. 1 (Jul., 1977), pp. 89-92.
https://www.jstor.org/stable/2041997

M. Hakim and N. Sibony, <i>Frontiere de Silov et spectre de A(D) pour des domaines faiblement pseudoconvexes,</i> C. R. Acad. Sci. Paris 281, Serie A (1975), 959-962.