  A $\Bbbk$-configuration of type $(d_1,\dots,d_s)$ is a specific set of points
in $\mathbb P^2$ that has a number of algebraic and geometric properties. For
example, the graded Betti numbers and Hilbert functions of all
$\Bbbk$-configurations in $\mathbb P^2$ are determined by the type
$(d_1,\dots,d_s)$. However the Waldschmidt constant of a $\Bbbk$-configuration
in $\mathbb P^2$ of the same type may vary. In this paper, we find that the
Waldschmidt constant of a $\Bbbk$-configuration in $\mathbb P^2$ of type
$(d_1,\dots,d_s)$ with $d_1\ge s\ge 1$ is $s$. We also find the Waldschmidt
constant of a standard $\Bbbk$-configuration in $\mathbb P^2$ of type $(a,b,c)$
with $a\ge 1$ except the type $(2,3,5)$. In particular, we prove that the
Waldschmidt constant of a standard $\Bbbk$-configuration in $\mathbb P^2$ of
type $(1,b,c)$ with $c\ge 2b+2$ does not depend on $c$.
