Let $\iota : S \hookrightarrow M$ be an inclusion that serves as an injective immersion between real manifolds of dimension $k$ and $n$, respectively. Fix $p \in S$. Then we have a linear embedding $\iota_{*, p} : T_{p}S \hookrightarrow T_{p}M$, and we wish to compute $\iota_{*, p}(T_{p}S)$. It is easy to see that $\iota_{*, p} (T_{p}S) \leqslant \{X_{p} \in T_{p}M : X_{p}(f) = 0 \text{ for all } f \in C^{\infty}(M) \text{ such that } f|_{S} = 0\}$.
Moreover, if $S$ is a regular submanifold (or embedded submanifold), meaning that we can take a chart $(U, x^{1}, \cdots, x^{n})$ near $p$ in $M$ so that $(U \cap S, x^{1}, \cdots, x^{k})$ is a chart near $p$ in $S$, one can prove that the above inclusion becomes equality.
I would like to know some examples where this equality does not hold. I imagine there must be some evident examples, but I have not succeeded in finding them. I appreciate your help.
CORRECTED:
If $S$ fails to be embedded in a neighborhood of $p\in M$, equality can fail. Consider a figure 8 in $\Bbb R^2\subset\Bbb R^3$ that is the image of a 1-1 immersion. Then if the two branches are tangent to the $x$- and $y$- axes at $p=0$, your right-hand side will be the $xy$-plane in $\Bbb R^3$. So this is an example where the right-hand side is strictly larger than $\iota_{*p}(T_pS)$ but strictly smaller than $T_pM$.