Contraction mapping in an incomplete metric space

Section 4 of this paper of Suzuki and Takahashi gives an example of a metric space -- in fact, a subspace of the Euclidean plane (so it seems you were on the right track!) -- which is incomplete but for which every contraction mapping has a fixed point.

They go on to repair matters by defining a "weakly contractive mapping" and showing that a metric space is complete iff every weakly contractive mapping has a fixed point.

Note: I was not aware of this paper until I read this question. I then googled -- contraction mapping, characterization of completeness -- and the paper showed up right away. (I look forward to reading it more carefully when I get the chance...)

There is a sense in which every such fixed point comes from a complete metric space:

Theorem: Let $X$ be an arbitrary set. Let $f:X\to X$ be a function with fixed point $x$ such that for all positive natural numbers $n$ and for all $y\neq x$, we have $f^n(y)\neq y$. Then there exists a complete metric $d$ on $X$ such that $f$ is a strict contraction.

The theorem is originally due to C. Bessaga and is not that easy to prove. A relatively short proof can be found here.