The fact that the rotation periods of Ap stars span five to six orders of magnitude and that the longest ones reach several hundred years represents one of the main unsolved challenges of stellar physics. Our goal is to gain better understanding of the occurrence and properties of the longest period Ap stars. We obtained high resolution spectra of a sample of super-slowly rotating Ap (ssrAp) star candidates identified by a TESS photometric survey to confirm that they are indeed Ap stars, to check that their projected equatorial velocities are compatible with super-slow rotation, and to obtain a first estimate of their magnetic field strengths. For the confirmed Ap stars, we determined whenever possible their mean magnetic field modulus, their mean quadratic magnetic field, and an upper limit of their projected equatorial velocities. Eighteen of the 27 stars studied are typical Ap stars; most of the other nine appear to be misclassified. One of the Ap stars is not a slow rotator; it must be seen nearly pole-on. The properties of the remaining 17 are compatible with moderately to extremely long rotation periods. Eight new stars with resolved magnetically split lines in the visible range were discovered; their mean magnetic field modulus and their mean quadratic magnetic field were measured. The mean quadratic field could also be determined in five more stars. Five spectroscopic binaries containing an Ap star, which were not previously known, were identified. Among the misclassified stars, one double-lined spectroscopic binary with two similar, sharp-lined Am components was also discovered. The technique that we used to carry out a search for ssrAp star candidates using TESS data is validated. Its main limitation appears to arise from uncertainties in the spectral classification of Ap stars. The new magnetic field measurements obtained as part of this study lend further support to the tentative conclusions of our previous studies: the absence of periods P_rot_>~150d in stars with B_0_>~7.5kG, the lower rate of occurrence of super-slow rotation for field strengths B_0_<~2kG than in the range ~3kG<B_0_<~7.5kG, and the deficiency of slowly rotating Ap stars with (phase-averaged) field strengths between ~2 and ~3kG.