  We discuss the response of vacuum fluctuations to a static potential in the
context of massive, ghost-free infinite-derivative scalar field theories in two
dimensions. For the special case of a $\delta$-like potential, $V=\lambda
\delta(x)$, the problem is exactly solvable and we calculate the corresponding
Hadamard function for this quantum field. Using this exact result we determine
the renormalized value of the vacuum polarization $\langle
\hat{\varphi}^2(x)\rangle_\text{ren}$ as a function of the distance $x$ from
the position of the potential. This expression depends on the amplitude of the
potential as well as the scale of non-locality $\ell$; for distances $x\gg\ell$
the non-local and local results agree, whereas for distances $x < \ell$ there
is a difference.
