We study a new graph separation problem called Multiway Near-Separator. Given an undirected graph $G$, integer $k$, and terminal set $T \subseteq V(G)$, it asks whether there is a vertex set $S \subseteq V(G) \setminus T$ of size at most $k$ such that in graph $G-S$, no pair of distinct terminals can be connected by two pairwise internally vertex-disjoint paths. Hence each terminal pair can be separated in $G-S$ by removing at most one vertex. The problem is therefore a generalization of (Node) Multiway Cut, which asks for a vertex set for which each terminal is in a different component of $G-S$. We develop a fixed-parameter tractable algorithm for Multiway Near-Separator running in time $2^{O(k \log k)} * n^{O(1)}$. Our algorithm is based on a new pushing lemma for solutions with respect to important separators, along with two problem-specific ingredients. The first is a polynomial-time subroutine to reduce the number of terminals in the instance to a polynomial in the solution size $k$ plus the size of a given suboptimal solution. The second is a polynomial-time algorithm that, given a graph $G$ and terminal set $T \subseteq V(G)$ along with a single vertex $x \in V(G)$ that forms a multiway near-separator, computes a 14-approximation for the problem of finding a multiway near-separator not containing $x$.