We present multiwavelength observations of the tidal disruption event (TDE) iPTF15af, discovered by the intermediate Palomar Transient Factory survey at redshift z=0.07897. The optical and ultraviolet (UV) light curves of the transient show a slow decay over 5 months, in agreement with previous optically discovered TDEs. It also has a comparable blackbody peak luminosity of L_peak_~1.5x10^44^erg/s. The inferred temperature from the optical and UV data shows a value of (3-5)x10^4^K. The transient is not detected in X-rays up to L_X_<3x10^42^erg/s within the first 5 months after discovery. The optical spectra exhibit two distinct broad emission lines in the HeII region, and at later times also H{alpha} emission. Additionally, emission from [NIII] and [OIII] is detected, likely produced by the Bowen fluorescence effect. UV spectra reveal broad emission and absorption lines associated with high-ionization states of NV, CIV, SiIV, and possibly PV. These features, analogous to those of broad absorption line quasars (BAL QSOs), require an absorber with column densities N_H_>10^23^cm^-2^. This optically thick gas would also explain the nondetection in soft X-rays. The profile of the absorption lines with the highest column density material at the largest velocity is opposite that of BAL QSOs. We suggest that radiation pressure generated by the TDE flare at early times could have provided the initial acceleration mechanism for this gas. Spectral UV line monitoring of future TDEs could test this proposal.