**EDIT:** Let $f\colon X\to Y$ be a morphism of complex analytic spaces (not necessarily smooth or reduced). Assume that

(a) $f$ is injective on points;

(b) $f$ is local imbedding near each point $x\in X$;

(c) for any holomorphic germ $\gamma\colon (\mathbb{C},0)\to Y$ with image contained in $f(X)$ there exists a holomorphic germ $\tilde\gamma\colon (\mathbb{C},0)\to X$ such that $f\circ \bar \gamma=\gamma$.

(d) $f(X)$ is a closed subset of $Y$.

**Question. Is it true that $f$ is proper?**

Remark. (1) In the algebraic situation the answer is positive by the standard valuative criterion of properness.

(2) A priori it is not clear to me that locally on $Y$ the image $f(X)$ is an analytic subset.

If there is any analytic version of the valuative criterion of properness, I would be curious to know anyway. A reference would be helpful.

**EDIT:** Conditions (a),(c), (d) imply the following property which looks closer to the standard valuative criterion of properness. For any holomorphic germs $\gamma\colon (\mathbb{C},0)\to Y$ and $\beta\colon (\mathbb{C},0)\backslash\{0\}\to X$ such that
$f\circ \beta=\gamma|_{(\mathbb{C},0)\backslash\{0\}}$, there exists a holomorphic germ $\bar\gamma\colon (\mathbb{C},0)\to X$ such that
$$\bar\gamma|_{(\mathbb{C},0)\backslash\{0\}}=\beta,\, f\circ \bar\gamma=\gamma.$$