  Given a function $f : A \to \mathbb{R}^n$ of a certain regularity defined on
some open subset $A \subseteq \mathbb{R}^m$, it is a classical problem of
analysis to investigate whether the function can be extended to all of
$\mathbb{R}^m$ in a certain regularity class. If an extension exists and is
continuous, then certainly it is uniquely determined on the closure of $A$. A
similar problem arises in general relativity for Lorentzian manifolds instead
of functions on $\mathbb{R}^m$. It is well-known, however, that even if the
extension of a Lorentzian manifold $(M,g)$ is analytic, various choices are in
general possible at the boundary. This paper establishes a uniqueness condition
for extensions of globally hyperbolic Lorentzian manifolds $(M,g)$ with a focus
on low regularities: any two extensions which are anchored by an inextendible
causal curve $\gamma : [-1,0) \to M$ in the sense that $\gamma$ has limit
points in both extensions, must agree locally around those limit points on the
boundary as long as the extensions are at least locally Lipschitz continuous.
We also show that this is sharp: anchored extensions which are only H\"older
continuous do in general not enjoy this local uniqueness result.
