[proofplan]
Work in a neighbourhood of $p$ on which the local complete intersection is cut out by the holomorphic functions $f_1,\ldots,f_r$. The ideal sheaf generated by these functions is the image of a morphism from a finite free $\mathcal{O}_X$-module, and the complete-intersection hypothesis gives a Koszul resolution of the quotient $\mathcal{O}_X/\mathcal{J}$. Since finite free analytic sheaves are coherent and coherence is preserved under kernels, images, and cokernels in finite exact sequences, the quotient sheaf, which is the structure sheaf $\mathcal{O}_V$, is coherent as an $\mathcal{O}_X$-module.
[/proofplan]
[step:Localize to the neighbourhood where the complete intersection is defined]
Choose an open neighbourhood $U \subset X$ of $p$ on which $V \cap U$ is defined by holomorphic functions $f_i:U \to \mathbb{C}$ for $1 \leq i \leq r$. Define the ideal sheaf $\mathcal{J} \subset \mathcal{O}_X|_U$ by
\begin{align*}
\mathcal{J}(W):=\sum_{i=1}^r f_i|_W\mathcal{O}_X(W)
\end{align*}
for every open set $W \subset U$. By definition of the analytic subspace cut out by $\mathcal{J}$,
\begin{align*}
\mathcal{O}_V|_U \cong \mathcal{O}_X|_U/\mathcal{J}.
\end{align*}
Since coherence is local, it is enough to prove that $\mathcal{O}_X|_U/\mathcal{J}$ is coherent.
[/step]
[step:Resolve the quotient by the Koszul complex of the defining equations]
Let $\mathcal{E}:=(\mathcal{O}_X|_U)^{\oplus r}$, and let $e_1,\ldots,e_r \in \mathcal{E}(U)$ denote its standard frame. Define the $\mathcal{O}_X|_U$-linear morphism
\begin{align*}
d_1:\mathcal{E} &\to \mathcal{O}_X|_U \\
\sum_{i=1}^r a_i e_i &\mapsto \sum_{i=1}^r a_i f_i,
\end{align*}
where each $a_i$ is a local section of $\mathcal{O}_X|_U$. Then $\operatorname{im}(d_1)=\mathcal{J}$.
For $0 \leq k \leq r$, define $\mathcal{K}_k:=\bigwedge^k_{\mathcal{O}_X|_U}\mathcal{E}$. The Koszul differential $d_k:\mathcal{K}_k \to \mathcal{K}_{k-1}$ is defined on $e_{i_1}\wedge\cdots\wedge e_{i_k}$ by
\begin{align*}
d_k(e_{i_1}\wedge\cdots\wedge e_{i_k})
=
\sum_{m=1}^k (-1)^{m-1}f_{i_m}
e_{i_1}\wedge\cdots\wedge \widehat{e_{i_m}}\wedge\cdots\wedge e_{i_k}.
\end{align*}
The local complete intersection hypothesis says that the germs $f_{1,q},\ldots,f_{r,q}$ form a regular sequence in $\mathcal{O}_{X,q}$ for $q \in V \cap U$. Hence, by the exactness of the Koszul complex for a regular sequence, the sequence
\begin{align*}
0 \to \mathcal{K}_r \xrightarrow{d_r} \mathcal{K}_{r-1} \to \cdots \to \mathcal{K}_1 \xrightarrow{d_1} \mathcal{O}_X|_U \to \mathcal{O}_X|_U/\mathcal{J} \to 0
\end{align*}
is exact.
[/step]
[step:Use that the Koszul terms are coherent]
The sheaf $\mathcal{O}_X|_U$ is coherent by the coherence of the analytic structure sheaf. Each $\mathcal{K}_k$ is finite locally free of rank $\binom{r}{k}$, hence locally isomorphic to a finite direct sum of copies of $\mathcal{O}_X|_U$. Since finite direct sums of coherent sheaves are coherent, each $\mathcal{K}_k$ is coherent.
[/step]
[step:Pass coherence to the quotient]
Let $\mathcal{Q}:=\mathcal{O}_X|_U/\mathcal{J}$. The Koszul resolution expresses $\mathcal{Q}$ as the cokernel of $d_1:\mathcal{K}_1 \to \mathcal{O}_X|_U$ inside a finite exact complex of coherent sheaves. By the stability of coherent sheaves under kernels and cokernels, applied successively to the exact Koszul complex, $\operatorname{im}(d_1)=\mathcal{J}$ is coherent and
\begin{align*}
\operatorname{coker}(d_1)\cong \mathcal{O}_X|_U/\mathcal{J}=\mathcal{Q}
\end{align*}
is coherent.
[/step]
[step:Identify the quotient with the structure sheaf]
By construction,
\begin{align*}
\mathcal{O}_V|_U \cong \mathcal{O}_X|_U/\mathcal{J}.
\end{align*}
The preceding step proves that the right-hand side is coherent as an $\mathcal{O}_X|_U$-module. Therefore $\mathcal{O}_V$ is coherent as an $\mathcal{O}_X$-module in a neighbourhood of $p$. Since the assertion is local, the structure sheaf of the local complete intersection is coherent.
[/step]
