As stated by Wikipedia, a mapping $f:X\rightarrow Y$ between metric spaces is locally Lipschitz continuous if for every $x\in X$ there exists a neighborhood $U$ of $x$ such that $f$ restricted to $U$ is Lipschitz continuous. If furthermore $X$ is locally compact, then $f$ is locally Lipschitz iff $f$ is Lipschitz continuous on every compact subset of $X$.
I observe that this statement and its proof were repeatedly asked and answered here. Also, a reference can be found in Exercise 4.2.10 in the book "Topology of Metric Spaces" by S. Kumaresan.
Besides that, may I ask for some other references for it?
Many thanks.