Can the Fermi surface be sharper than k_BT?

We consider a "metal" with a mean free path possessing a sharp maximum at the Fermi surface. At finite temperature, the width of this maximum is close to k_BT. We show that the electrical conductivity of this system behaves as if the peak of the mean free path remains sharp at finite temperatures. In contrast to the well-known normal situation, the resistivity due to elastic scattering is surprisingly found to be linear in temperature, and the resistivity due to scattering by phonons is proportional to T². This model is proposed to relate to some properties of "doped insulators", such as cuprates, organic metals and fullerenes.