Let $X$ be a nonsingular variety over an algebraically closed field and $x$ be a closed point on $X$.
Then one defines the tangent space in $x$ as the $k-$ vector space
$T_x(X):= \operatorname{Hom}_k(\mathfrak m / \mathfrak m^2, k)$,
where $\mathfrak m$ is the maximal ideal of the point. Now consider the $k-$vector space $\mathfrak m / \mathfrak m^2$: you can also see it as just an $\mathcal O_{X,x}$ - module.
Now the question: under the above hypotheses: is $\mathfrak m / \mathfrak m^2$ already as such $\mathcal O_{X,x}$ - module free?