Suppose we are given a continuous path,
$$\gamma:[0,1]\rightarrow (X,d)\text{,}$$
in a metric space $(X,d)$. When we deal with differentiable enough paths in Riemann manifolds we can give a parametrization that is $l$-Lipschitz, where $l$ is the length of the path. If have that $\gamma$ has finite length $l$ in the sense that $$l=\sup\left\{\sum_{i=0}^{n-1}d(\gamma(t_i),\gamma(t_{i+1}))\,|\, n\in \mathbb{N}; \forall i< n,t_i\in[0,1]\text{ and }t_i<t_{i+1};t_0=0;t_n=1\right\}$$ is finite, can we guarantee the existence of a parametrization that makes the path $l$-Lipschitz?
Note: The necessity for this result comes that I need this fact to prove that certain sequences of path in metric spaces have nice properties in order to obtain a limit path. And in this way proving that compact path metric spaces are geodesic metric spaces.
Yes, every rectifiable path can be parametrized by arclength (and a locally rectifiable path too, if you allow the parameter domain to be an unbounded interval).
Let $L(x)$ be the length of $\gamma_{|[0,x]}$. This is a nondecreasing function of $x$. It is constant precisely on the intervals where $\gamma$ is constant. Thus, $\gamma\circ L^{-1}$ is well defined. This is the desired parametrization.