Large-scale $in\ vivo$ electrophysiology requires tools that enable simultaneous recording of multiple brain regions at single-neuron level. This calls for the design of more compact neural probes that offer even larger arrays of addressable sites and high channel counts. With this aim, we present in this paper a quad-shank approach to integrate as many as 5,120 sites on a single probe. Compact fully-differential recording channels were designed using a single-gain-stage neural amplifier with a 14-bit ADC, achieving a mean input-referred noise of 7.44 μ $\text{V}_{\text{rms}}$ in the action-potential band and 7.65 μ $\text{V}_{\text{rms}}$ in the local-field-potential band, a mean total harmonic distortion of 0.17% at 1 kHz and a mean input-referred offset of 169 μV. The probe base incorporates 384 channels with on-chip power management, reference-voltage generation and digital control, thus achieving the highest level of integration in a neural probe and excellent channel-to-channel uniformity. Therefore, no calibration or external circuitry are required to achieve the above-mentioned performance. With a total area of $2.2 \times 8.67\text{ mm}^2$ and a power consumption of 36.5 mW, the presented probe enables full-system miniaturization for acute or chronic use in small rodents.