Does sodium produce water when it reacts with water?
The reaction of sodium metal and water could (and usually does) indirectly produce more water, if enough heat is produced to spontaneously ignite the hydrogen gas which is produced by the initial reaction. This cyclic type of reaction is often found in explosive or extremely exothermic reactions, e.g. $\ce{K+ + I- + H2O2}$ will also react violently in a cyclic way.

Combustion of hydrogen in air is this: 

$$\ce{2H2 + O2 -&gt; 2H2O}$$

So, in a (well) closed system with excess oxygen, water could re-form and collect. It's worth noting that dissolution of $\ce{NaOH}$ in water ($\ce{NaOH -&gt; Na+ + OH-}$) is also exothermic and adds even more heat to the system.