It's a rather easy exercise to construct a non-complete metric space $X$ such that any continuous mapping $f:X\to X$ has a fixed point. However, I'd like to have a reference to such an example.
For instance, section 4 of the paper
Suzuki, Tomonari; Takahashi, Wataru. Fixed point theorems and characterizations of metric completeness. Topol. Methods Nonlinear Anal. 8 (1996), no. 2, 371–382 (1997); MR1483635
contains such an example.
For more information see Converse to Banach's fixed point theorem? at MathOverflow, Contraction mapping in an incomplete metric space at math.SE,
and related pages pointed out at these sites.