Prefix/suffix property of language containing only empty word
You probably mean $L=\{\varepsilon\}$, i.e. the language which contains only the empty word, as opposed to the empty language. Both the empty language and the language which contains only the empty word have the prefix/suffix property, since they satisfy the definion.

However, the prefix property is a property of a code, not a property of a language. A language can describe a code, but neither the empty language nor the language which contains only the empty word describe a code. So in the end it is irrelevant whether those languages have the prefix/suffix property.

Edit rici is right that Hopcroft&amp;Ullman say that the LR(0) grammars define exactly the DCFL's having the prefix property. (They hasten to add that the prefix property is not a severe restriction.) So in this case it is good that the empty language and the language which contains only the empty word have the prefix property. Additionally, Berstel&amp;Perrin allow the empty set as a code, so you can also dispute my other claim. But they agree that $\{\varepsilon\}$ has the prefix property, and is not a code:


  Proposition 2.1.9 Any prefix (suffix, bifix) set of words $X\neq \{\varepsilon\}$ is a code.


Both Hopcroft&amp;Ullman and Berstel&amp;Perrin consider $\varepsilon$ to be a prefix/suffix of any word, but they require a proper prefix/suffix in the definition of the prefix/suffix property. (This is clearly the right thing to do, otherwise no set of words would have the prefix/suffix property.)