Broad absorption lines (BALs) found in a significant fraction of quasar spectra identify high-velocity outflows that might be present in all quasars and could be a major factor in feedback to galaxy evolution. Understanding the nature of these flows requires further constraints on their physical properties, including their column densities, for which well-studied BALs, such as C iv λλ1548,1551, typically provide only a lower limit because of saturation effects. Low-abundance lines, such as P v λλ1118,1128, indicate large column densities, implying that outflows more powerful than measurements of C iv alone would indicate. We search through a sample of 2694 BAL quasars from the Sloan Digital Sky Survey III/Baryon Oscillation Spectroscopic Survey data release 9 quasar catalogue for such absorption, and we identify 81 'definite' and 86 'probable' detections of P v broad absorption, yielding a firm lower limit of 3.0-6.2 per cent for the incidence of such absorption among BAL quasars. The P v-detected quasars tend to have stronger C iv and Si iv absorption, as well as a higher incidence of LoBAL absorption, than the overall BAL quasar population. Many of the P v-detected quasars have C iv troughs that do not reach zero intensity (at velocities where P v is detected), confirming that the outflow gas only partially covers the UV continuum source. P v appears significantly in a composite spectrum of non-P v-detected BAL quasars, indicating that P v absorption (and large column densities) is much more common than indicated by our search results. Our sample of P v detections significantly increases the number of known P v detections, providing opportunities for follow-up studies to better understand BAL outflow energetics.