  We study the response of a scalar thermal field to a $\delta$-probe in the
context of non-local ghost-free theories. In these theories a non-local form
factor is inserted into the kinetic part of the action which does not introduce
new poles. For the case of a static $\delta$-potential we obtain an explicit
expression for the thermal Hadamard function and use it for the calculation of
the thermal fluctuations. We then demonstrate how the presence of non-locality
modifies the amplitude of these fluctuations. Finally, we also discuss the
fluctuation-dissipation theorem in the context of ghost-free quantum field
theories at finite temperature.
