We present observations of the most radio-luminous broad absorption-line (BAL) quasar known, 1624+3758, at redshift z = 3.377. The quasar has several unusual properties. (1) The Fe II UV191 1787-A emission line is very prominent. (2) The BAL trough (BALnicity index 2990 km s −1 )i sdetached by 21 000 km s −1 and extends to velocity v =− 29 000 km s −1 . There are additional intrinsic absorbers at −1900 and −2800 km s −1 . (3) The radio rotation measure of the quasar, 18 350 rad m −2 ,i sthe second highest known. The radio luminosity is P 1.4 GHz = 4.3 × 10 27 WH z −1 (H 0 = 50 km s −1 Mpc −1 , q 0 = 0.5) and the radio loudness is R ∗ = 260. The radio source is compact ( ∼ 2.8 kpc) and the radio spectrum is GHz-peaked, consistent with it being relatively young. The width of the C IV emission line, in conjunction with the total optical luminosity, implies a black hole mass M BH ∼ 10 9 M� , L/L Eddington ≈ 2. The high Eddington ratio and the radio-loudness place this quasar in one corner of Boroson’s two-component scheme for the classification of active galactic nuclei, implying a very high accretion rate, and this may account for some of the unusual observed properties. The v = −1900 km s −1 absorber is a possible Lyman-limit system, with N (H I) = 4 × 10 18 cm −2 , and a covering factor of 0.7. A complex mini-BAL absorber at v =− 2200 to −3400 km s −1 is detected in each of C IV ,N V and O VI. The blue and red components of the C IV doublet happen to be unblended, allowing both the covering factor and optical depth to be determined as a function of velocity. Variation of the covering factor with velocity dominates the form of the mini-BAL, with the absorption being saturated (e −τ ≈ 0) over most of the velocity range. The velocity dependence of the covering factor and the large velocity width imply that the mini-BAL is intrinsic to the quasar. There is some evidence of line-locking between velocity components in the C IV mini-BAL, suggesting that radiation pressure plays a role in accelerating the outflow. Ke yw ords: galaxies: high-redshift ‐ intergalactic medium ‐ quasars: absorption lines ‐ quasars: emission lines ‐ quasars: general ‐ early Universe.