It seems that I have the needed example, but I want it to be simple and self-explaining...

Construct a nontrivial complete metric space $X$ with intrinsic metric which has no nontrivial minimizing geodesics.

**Definitions:**

A metric $d$ is called

*intrinsic*if for any two points $x$, $y$ and any $\epsilon>0$ there is an $\epsilon$-midpoint $z$; i.e. $d(x,z),d(z,y)<\tfrac12 d(x,y)+\epsilon$.A minimizing geodesic is

*nontrivial*if it connects two distinct points.A meric space is

*nontrivial*if it contains two distinct points.

**Comments:**

- Clearly, $X$ can not be locally compact.