Positron lifetime spectra for multiwall carbon nanotubes doped with the alkali metals, K, Rb and Cs, are found to consist of a single component with a lifetime of 0.36 ns at 280 K, independent of the alkali metal. The shorter lifetime than in pristine carbon nanotubes (0.382 ns) demonstrates positron annihilation at a site of higher electron density. We conclude that all positrons injected into alkali metal-doped carbon nanotubes annihilate at positron-trapping centers, derived probably from extended lattice defects located on their negatively-charged graphitic shells.