Assume that $ \omega$ is a selective nonprinciple ultrafilter. Fix $\omega$.
Def : Define $X_\omega$ to be set of equivalence classes of sequence $(x_n),\ x_n\in X_n$ where $X_n$ is a metric space. And we have a metric $d_\omega$ on $X_\omega$ $$ (x_n)\sim (y_n)\Leftrightarrow \lim_{n\rightarrow \omega}\ |x_n-y_n|=0 $$
Question : Consider $$ X'=\bigcup_{i=1}^\infty \ \bigg[0,1+ \frac{1}{i}\bigg] $$ where union is disjoint and each segment $[0,1+\frac{1}{i}]$ has an Euclidean metric. And $X$ is quotient space of $X'$ by identifying $0\sim p$ and $1+ \frac{1}{i} \sim q$.
That is $X$ is a complete length space and not locally compact.
Then limit of $[0,1+\frac{1}{n}] \subset X_n:=X$ is a geodesic of length $1$ between $p$ and $q$ in $X_\omega$.
What is another geodesic between $p$ and $q$ in $X_\omega$ ? Thank you in advance.
For every sequence $s=(i_n)$ whose limit (in the sense of your ultrafilter $\omega$) is $\infty$, the limit $$ \lim_\omega ~[0, i_n] $$ is a geodesic $g_s$ connecting $p$ to $q$ in $X_\omega$. Moreover, $g_s=g_t$ if and only if the sequences $s$ and $t$ determine the same element of the ultrapower $${\mathbb N}^*=\left(\prod_{i\in {\mathbb N}}{\mathbb N}\right)/\omega$$ of ${\mathbb N}$. The latter has cardinality of continuum. As for "examples" other than the obvious one ($i_n=n$), good luck with that.