Does every connected metrizable locally path connected topological space $X$ admit a compatible metric $d$ so that $(X,d)$ is a length space?

(**Edit to correct definition:** Recall that a metric space $(X,d)$ is a length space if for every $x$ and $y$ in $X$ and every $e>0$, there exists a rectifiable path from $x$ to $y$ whose length is less then $d(x,y)+e$.)

The answer is certainly *yes* for Peano continua, but this is not a trivial fact.

More generally the answer is apparently *yes* for such locally compact spaces, but local compactness is certainly not necessary: for example, familiar Hilbert space is a length space.

Do the above claims survive without local compactness?