  In this work we show that for the geodesic spray $S$ of a Finsler function
$F$ the most natural projective deformation $\widetilde{S}=S -2 \lambda
F\mathbb C$ leads to a non-Finsler metrizable spray, for almost every value of
$\lambda \in \mathbb R$. This result shows how rigid is the metrizablility
property with respect to certain reparameterizations of the geodesics. As a
consequence we obtain that the projective class of an arbitrary spray contains
infinitely many sprays that are not Finsler metrizable.
