We report on the discovery of a peculiar broad absorption line (BAL) quasar identified in our Gaia-assisted survey of red quasars. The systemic redshift of this quasar was difficult to establish because of the absence of conspicuous emission lines. Based on deep and broad BAL troughs of at least Si IV, C IV, and Al III, a redshift of z = 2.41 was established under the assumption that the systemic redshift can be inferred from the red edge of the BAL troughs. However, we observe a weak and spatially extended emission line at 4450 Å that is most likely due to Lyman-α emission, which implies a systemic redshift of z = 2.66 if correctly identified. There is also evidence for the onset of Lyman-α forest absorption bluewards of 4450 Å and evidence for Hα emission in the K band consistent with a systemic redshift of z = 2.66. If this redshift is correct, the quasar is an extreme example of a detached low-ionisation BAL quasar. The BALs must originate from material moving with very large velocities ranging from 22 000 km s-1 to 40 000 km s-1. To our knowledge, this is the first case of a systemic-redshift measurement based on extended Lyman-α emission for a BAL quasar. This method could also be useful in cases of sufficiently distant BL Lac quasars without systemic-redshift information.