TOI-1416 (BD+42 2504, HIP 70705) is a V=10 late G- or early K-type dwarf star. TESS detected transits in its Sectors 16, 23, and 50 with a depth of about 455ppm and a period of 1.07 days. Radial velocities (RVs) confirm the presence of the transiting planet TOI-1416 b, which has a mass of 3.48+/-0.47M_{Earth}_ and a radius of 1.62+/-0.08R_{Earth}_, implying a slightly sub-Earth density of 4.50^+0.90^_-0.83_g/cm^3^. The RV data also further indicate a tentative planet, c, with a period of 27.4 or 29.5 days, whose nature cannot be verified due to strong suspicions of contamination by a signal related to the Moon's synodic period of 29.53 days. The nearly ultra-short-period planet TOI-1416 b is a typical representative of a short-period and hot (Teq=~1570K) super-Earth-like planet. A planet model of an interior of molten magma containing a significant fraction of dissolved water provides a plausible explanation for its composition, and its atmosphere could be suitable for transmission spectroscopy with JWST. The position of TOI-1416 b within the radius-period distribution corroborates the idea that planets with periods of less than one day do not form any special group. It instead implies that ultra-short-period planets belong to a continuous distribution of super-Earth-like planets with periods ranging from the shortest known ones up to =~30 days; their period-radius distribution is delimited against larger radii by the Neptune Desert and by the period-radius valley that separates super-Earths from sub-Neptune planets. In the abundance of small, short-periodic planets, a notable plateau has emerged between periods of 0.6-1.4 days, which is compatible with the low-eccentricity formation channel. For the Neptune Desert, its lower limits required a revision due to the increasing population of short-period planets; for periods shorter then 2 days, we establish a radius of 1.6R_{Earth}_ and a mass of 0.028M_jup_ (corresponding to 8.9M_{Earth}_) as the desert's lower limits. We also provide corresponding limits to the Neptune Desert against the planets' insolation and effective temperatures.