The core-excited quartet spectrum of Na i produced by beam-foil excitation has been studied in the 3300--5500-A\r{} region. Sixteen new lines, representing transitions between the 2${p}^{5}$3s3p, 2${p}^{5}$3s3d, and 2${p}^{5}$3s4s configurations, are reported, together with the lifetimes for 12 of the levels belonging to the 2${p}^{5}$3s3d and 2${p}^{5}$3s4s configurations. Some of the lifetimes deviate considerably from recently published values [L. Engstr\"om, L. Young, L. P. Sommerville, and H. Berry, Phys. Rev. A 32, 1468 (1985)], including the lifetimes of the 2${p}^{5}$3s3d $^{4}\mathrm{D}_{3/2}$ and $^{4}\mathrm{F}_{5/2}$ levels, which have both been suggested for use in an extreme ultraviolet (xuv) laser. The experimental data for fine-structure splittings and lifetimes are in good agreement with multiconfiguration Hartree-Fock calculations.