Can a battleship float in a tiny amount of water?

Yes it floats. And it has displaced its "own weight of water" in the sense that if you had filled the container with water and only then lowered the ship into the container, nearly all that water would have been dispaced and is now sloshing around on the floor.

Note that the water does not need to have been present - this calculation gives just the way to calculate the non water volume occupied by the floating object (that hence is unavailable for water).

So in your example, assume you put the ship into the container. If you filled it up to the same water level without the ship being present - that is the displacement caused by the ship.

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wBBBw     wwwww     w   w
wBBBw vs. wwwww NOT w   w
wwwww     wwwww     wwwww


In this sketch, 6 w have been "displaced".

Hence your example works even with a tiny gap. You can argue that hydraulic machines work with 0 gaps. The piston is the boat in the cylinder.

The USS Missouri $$5.8 \times 10^7\,\rm kg, \, 270\,\rm m$$ long with a fully laden draft of $$11.5\,\rm m$$ has an underwater surface area in excess of $$270\times 11.5\times 2 \approx 6200\,\rm m^2$$ and needs to "displace" $$5.8 \times 10^7\,\rm kg$$ of salt water (density $$\approx 1020 \,\rm kg \, m^{-3}$$) to float.

Assume a custom made tank so that an even thickness of water (total volume $$1 \,\rm litre = 0.001 \,m^{-3}$$) surrounds the USS Missouri below its waterline.
This thickness of the water layer would be smaller than $$\frac{0.001}{6200} \approx 1.6 \times 10^{-7} \rm m$$.

So in theory possible but in practice very highly unlikely.

The OP has changed the title from "1 litre" to "a small amount of water".
All that needs to be done is to choose a volume of water such that it is practically possible to float the USS Missouri in a suitably shaped dock and the OP's layer of water 3 cm thick might be possible in practice?

The picture in this answer gives a flavour of an "apparent lack of water" being able to float a ship.