Projective line with a doubled point
= Projective line with a doubled point
Glue two copies of $\mathbb P_k^1$ by the identity away from one chosen point. The resulting finite-type scheme is universally closed over $k$: after any base change, a closed subset meets each projective-line chart in a closed subset whose image is closed, and the total image is their finite union. The two copies of the chosen point cannot be separated, so the structure morphism is not separated and therefore is not proper.