Suppose that the lemma holds whenever $Y$ is a scheme. Let $U$ be a scheme and let $U \to Y$ be a surjective \'etale morphism. Let $R = U \times_Y U$ with projections $t, s : R \to U$. Denote $X_U = U \times_Y X$ and $\mathcal{L}_U$ the pullback. Then we get an open subscheme $V' \subset U$ as in the lemma for $(X_U \to U, \mathcal{L}_U)$. By the functorial characterization we see that $s^{-1}(V') = t^{-1}(V')$. Thus there is an open subspace $V \subset Y$ such that $V'$ is the inverse image of $V$ in $U$. In particular $V' \to V$ is surjective \'etale and we conclude that $\mathcal{L}_V$ is ample on $X_V/V$ (Divisors on Spaces, Lemma \ref{spaces-divisors-lemma-relatively-ample-local}). Now, if $Y' \to Y$ is a morphism such that $\mathcal{L}'$ is ample on $X'/Y'$, then $U \times_Y Y' \to Y'$ must factor through $V'$ and we conclude that $Y' \to Y$ factors through $V$. Hence $V \subset Y$ is as in the statement of the lemma. In this way we reduce to the case dealt with in the next paragraph. \medskip\noindent Assume $Y$ is a scheme. Since the question is local on $Y$ we may assume $Y$ is an affine scheme. We will show the following: \begin{enumerate} \item[(A)] If $\Spec(k) \to Y$ is a morphism such that $\mathcal{L}_k$ is ample on $X_k/k$, then there is an open neighbourhood $V \subset Y$ of the image of $\Spec(k) \to Y$ such that $\mathcal{L}_V$ is ample on $X_V/V$. \end{enumerate} It is clear that (A) implies the truth of the lemma. \medskip\noindent Let $X \to Y$, $\mathcal{L}$, $\Spec(k) \to Y$ be as in (A). By Lemma \ref{lemma-descending-property-ample} we may assume that $k = \kappa(y)$ is the residue field of a point $y$ of $Y$. \medskip\noindent As $Y$ is affine we can find a directed set $I$ and an inverse system of morphisms $X_i \to Y_i$ of algebraic spaces with $Y_i$ of finite presentation over $\mathbf{Z}$, with affine transition morphisms $X_i \to X_{i'}$ and $Y_i \to Y_{i'}$, with $X_i \to Y_i$ proper and of finite presentation, and such that $X \to Y = \lim (X_i \to Y_i)$. See Limits of Spaces, Lemma \ref{spaces-limits-lemma-proper-limit-of-proper-finite-presentation-noetherian}. After shrinking $I$ we may assume $Y_i$ is an (affine) scheme for all $i$, see Limits of Spaces, Lemma \ref{spaces-limits-lemma-limit-is-affine}. After shrinking $I$ we can assume we have a compatible system of invertible $\mathcal{O}_{X_i}$-modules $\mathcal{L}_i$ pulling back to $\mathcal{L}$, see Limits of Spaces, Lemma \ref{spaces-limits-lemma-descend-invertible-modules}. Let $y_i \in Y_i$ be the image of $y$. Then $\kappa(y) = \colim \kappa(y_i)$. Hence $X_y = \lim X_{i, y_i}$ and after shrinking $I$ we may assume $X_{i, y_i}$ is a scheme for all $i$, see Limits of Spaces, Lemma \ref{spaces-limits-lemma-limit-is-scheme}. Hence for some $i$ we have $\mathcal{L}_{i, y_i}$ is ample on $X_{i, y_i}$ by Limits, Lemma \ref{limits-lemma-limit-ample}. By Divisors on Spaces, Lemma \ref{spaces-divisors-lemma-ample-in-neighbourhood} we find an open neigbourhood $V_i \subset Y_i$ of $y_i$ such that $\mathcal{L}_i$ restricted to $f_i^{-1}(V_i)$ is ample relative to $V_i$. Letting $V \subset Y$ be the inverse image of $V_i$ finishes the proof (hints: use Morphisms, Lemma \ref{morphisms-lemma-ample-base-change} and the fact that $X \to Y \times_{Y_i} X_i$ is affine and the fact that the pullback of an ample invertible sheaf by an affine morphism is ample by Morphisms, Lemma \ref{morphisms-lemma-pullback-ample-tensor-relatively-ample}).