Interventional applications of photoacoustic imaging typically require visualization of point-like targets, such as the small, circular, cross-sectional tips of needles, catheters, or brachytherapy seeds. When these point-like targets are imaged in the presence of highly echogenic structures, the resulting photoacoustic wave creates a reflection artifact that may appear as a true signal. We propose to use deep learning techniques to identify these types of noise artifacts for removal in experimental photoacoustic data. To achieve this goal, a convolutional neural network (CNN) was first trained to locate and classify sources and artifacts in pre-beamformed data simulated with -Wave. Simulations initially contained one source and one artifact with various medium sound speeds and 2-D target locations. Based on 3,468 test images, we achieved a 100% success rate in classifying both sources and artifacts. After adding noise to assess potential performance in more realistic imaging environments, we achieved at least 98% success rates for channel signal-to-noise ratios (SNRs) of -9dB or greater, with a severe decrease in performance below -21dB channel SNR. We then explored training with multiple sources and two types of acoustic receivers and achieved similar success with detecting point sources. Networks trained with simulated data were then transferred to experimental waterbath and phantom data with 100% and 96.67% source classification accuracy, respectively (particularly when networks were tested at depths that were included during training). The corresponding mean ± one standard deviation of the point source location error was 0.40 ± 0.22 mm and 0.38 ± 0.25 mm for waterbath and phantom experimental data, respectively, which provides some indication of the resolution limits of our new CNN-based imaging system. We finally show that the CNN-based information can be displayed in a novel artifact-free image format, enabling us to effectively remove reflection artifacts from photoacoustic images, which is not possible with traditional geometry-based beamforming.