Optical coherent tomography (OCT) fingerprint technology provides rich depth information, including internal fingerprint (papillary junction) and sweat (eccrine) glands, in addition to imaging any fake layers (presentation attacks) placed over finger skin. Unlike 2D surface fingerprint scans, additional depth information provided by the cross-sectional OCT depth profile scans are purported to thwart fingerprint presentation attacks. We develop and evaluate a presentation attack detector (PAD) based on deep convolutional neural network (CNN). Input data to CNN are local patches extracted from the cross-sectional OCT depth profile scans captured using THORLabs Telesto series spectral-domain fingerprint reader. The proposed approach achieves a TDR of 99.73% @ FDR of 0.2% on a database of 3,413 bonafide and 357 PA OCT scans, fabricated using 8 different PA materials. By employing a visualization technique, known as CNN-Fixations, we are able to identify the regions in the OCT scan patches that are crucial for fingerprint PAD detection.