Exactly one of these statements is false. Which one? (a) When ammonia dissolves in water, the latter behaves as a Lewis base.
(b) The conjugate acid of $\ce$ is $\ce.$
(c) Boric acid is a monobasic acid.
(d) Ammonia behaves as an acid when it reacts with sodium metal to give sodamide and hydrogen.
The key says (d) is the false statements. How? I think (a) is the one. (d) $\ce <2NH3 + 2Na ->2NaNH2 + H2>$ The ammonia here loses a proton, which is characteristic for acids. In addition, it attaches to the electropositive sodium. (a) On the other hand, when ammonia dissolves in water: $\ce NH4+ + OH->$ Ammonia gains a proton, which is characteristic for bases.
Your reasoning is correct. Perhaps the answer key is mistaken.
For part (a), in the reaction
water is donating a proton $(\ce$ ion) and hence is behaving as a Brønsted acid. Since all Brønsted acids are Lewis acids, water is behaving as a Lewis acid.
For part (d) as well, the half reactions are:
It's clear that $\ce$ is acting as an acid and $\ce$ is acting as a Lewis base by donating electrons. The $\ce$ ion pairs up with the $\ce$ ion to form $\ce$ and hydrogen accepts the electrons and is liberated as $\ce.$
EDIT: As pointed out by Mithoron in the comments, the reduction of ammonia is not so simple. Sodium dissolves in ammonia to give a deep-blue solution containing ammoniated electrons (reference here):
The $\ce$ eventually loses a hydrogen atom to give $\ce$ , which then combines with the $\ce$ giving the product.