Is it a prefix code?

Pyth, 8 bytes

.AxM.PQ2

Test suite

Take all 2 element permutations of the input, map each to the index of one string in the other (0 for a prefix) and return whether all results are truthy (non-zero).

Haskell, 37 bytes

f l=[x|x<-l,y<-l,zip x x==zip x y]==l

Each element x of l is repeated once for every element that it's a prefix of, which is exactly once for a prefix-free list, giving the original list. The prefix property is checked by zipping both lists with x, which cuts off elements beyond the length of x.

Java, 128 127 126 125 124 121 bytes

(Thanks @Kenny Lau, @Maltysen, @Patrick Roberts, @Joba)

Object a(String[]a){for(int i=0,j,l=a.length;i<l;i++)for(j=0;j<l;)if(i!=j&a[j++].startsWith(a[i]))return 1<0;return 1>0;}

Ungolfed

Object a(String[] a) {
    for (int i = 0, j, l = a.length; i < l; i++) 
        for (j = 0; j < l;) 
            if (i != j & a[j++].startsWith(a[i])) return 1<0;
    return 1>0;
}

Output

[Hello, World]
true

[Code, Golf, Is, Cool]
true

[1, 2, 3, 4, 5]
true

[This, test, case, is, true]
true

[111, 010, 000, 1101, 1010, 1000, 0111, 0010, 1011, 0110, 11001, 00110, 10011, 11000, 00111, 10010]
true

[4, 42]
false

[1, 2, 3, 34]
false

[This, test, case, is, false, t]
false

[He, said, Hello]
false

[0, 00, 00001]
false

[Duplicate, Duplicate, Keys, Keys]
false