Given is a connected graph with undirected edges such that deleting any vertex decomposes the graph into $\le k$ components, for some $k\ge 2$. Is it true that the graph has a spanning tree with degree at most $k$?

The claim is clearly true if the graph itself is a tree. I checked some articles on vertex connectivity but couldn't find anything related to this.

Edit: Tony Huynh found a counterexample. What if the graph is required to be such that removing some vertex does not give a connected graph?