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We present statistical analysis of 11,200 proton kinetic-scale current sheets (CS) observed by Parker Solar Probe during 10 days around the first perihelion. The CS thickness $λ$ is in the range from a few to 200 km with the typical value around 30 km, while current densities are in the range from 0.1 to 10\;$μ{\rm A/m^2}$ with the typical value around 0.7\;$μ{\rm A/m^2}$. These CSs are resolved thanks to magnetic field measurements at 73--290 Samples/s resolution. In terms of proton inertial length $λ_{p}$, the CS thickness $λ$ is in the range from about $0.1$ to $10λ_{p}$ with the typical value around 2$λ_{p}$. The magnetic field magnitude does not substantially vary across the CSs and, accordingly, the current density is dominated by the magnetic field-aligned component. The CSs are typically asymmetric with statistically different magnetic field magnitudes at the CS boundaries. The current density is larger for smaller-scale CSs, $J_0\approx 0.15 \cdot (λ/100\;{\rm km})^{-0.76}$ $μ{\rm A/m^2}$, but does not statistically exceed the Alfvén current density $J_A$ corresponding to the ion-electron drift of local Alfvén speed. The CSs exhibit remarkable scale-dependent current density and magnetic shear angles, $J_0/J_{A}\approx 0.17\cdot (λ/λ_{p})^{-0.67}$ and $Δθ\approx 21^{\circ}\cdot (λ/λ_{p})^{0.32}$. Based on these observations and comparison to recent studies at 1 AU, we conclude that proton kinetic-scale CSs in the near-Sun solar wind are produced by turbulence cascade and they are automatically in the parameter range, where reconnection is not suppressed by the diamagnetic mechanism, due to their geometry dictated by turbulence cascade.