Precise Point Positioning (PPP), a cutting edge GNSS technology, can achieve high-precision positioning without base station assistance. Visual-Inertial Odometry (VIO) realizes a more robust local pose estimation than Visual-SLAM. Based on PPP and VIO, we propose a tightly-coupled PPP/INS/Visual SLAM system, P$^{3}$-VINS. It fuses GNSS raw measurements (pseudorange, carrier phase, and Doppler) with visual and inertial information for accurate and robust state estimation. All raw data is modelled and optimized under a factor graph framework. To eliminate ionospheric effects and utilize carrier phase measurements, P$^{3}$-VINS uses the ionosphere-free (IF) model by dual-frequency observations and adds phase ambiguity into the estimated states. Finally, P$^{3}$-VINS is evaluated on both public datasets and real-world experiments. It significantly outperforms benchmarks (GVINS and PPP) in terms of accuracy and smoothness. This result demonstrates that the high precision carrier phase substantially helps the GNSS/INS/Visual SLAM system reduce noise and improve accuracy.