  We derive an exact formula for the field form factor in the anyonic
Lieb-Liniger model, valid for arbitrary values of the interaction $c$, anyonic
parameter $\kappa$, and number of particles $N$. Analogously to the bosonic
case, the form factor is expressed in terms of the determinant of a $N\times N$
matrix, whose elements are rational functions of the Bethe quasimomenta but
explicitly depend on $\kappa$. The formula is efficient to evaluate, and
provide an essential ingredient for several numerical and analytical
calculations. Its derivation consists of three steps. First, we show that the
anyonic form factor is equal to the bosonic one between two special off-shell
Bethe states, in the standard Lieb-Liniger model. Second, we characterize its
analytic properties and provide a set of conditions that uniquely specify it.
Finally, we show that our determinant formula satisfies these conditions.
