False:

Explanation:
- In a vertically submerged plane surface, the pressure at every point does not remain the same.
- The pressure at a point in a fluid is given by the equation P = P₀ + ρgh, where P is the pressure at the point, P₀ is the pressure at the free surface of the fluid, ρ is the density of the fluid, g is the acceleration due to gravity, and h is the depth of the point from the free surface.
- As the depth of the point from the free surface changes, the pressure at that point also changes.
- The pressure increases with an increase in depth. This is because the weight of the fluid column above the point increases with depth, resulting in an increase in pressure.
- Similarly, the pressure decreases with a decrease in depth.
- Therefore, the pressure at different points in a vertically submerged plane surface will vary depending on their depth from the free surface.
- This variation in pressure is important in various applications, such as calculating the hydrostatic forces on submerged structures or determining the force exerted by a fluid on a submerged object.
- It is essential to consider the variation in pressure at different points when analyzing and designing structures that interact with fluids, as it affects the stability and behavior of these structures.

Conclusion:
- The statement "In a vertically submerged plane surface, the pressure at every point remains the same" is false. The pressure at different points in a vertically submerged plane surface varies depending on their depth from the free surface.