Let $A$ denote a sequence of bits.
Let $H$ denote a cryptographic hash function that has no limit on the length of its input (for example, SHA-3).
Consider the following infinite sequence of bits: $$B = H(A) \mathbin\Vert H(0 \mathbin\Vert A) \mathbin\Vert H(1 \mathbin\Vert A) \mathbin\Vert H(00 \mathbin\Vert A) \mathbin\Vert H(01 \mathbin\Vert A) \mathbin\Vert H(10 \mathbin\Vert A) \mathbin\Vert \ldots$$ (that is, the prefix goes through all possible bitstrings, sorted by their lengths).
Can we assume that for any chosen $A$, its corresponding $B$ is unique and contains all possible finite bitstrings (similarly to a binary representation of a fractional part of Pi)? If yes, can we use such technique as a cryptographically secure pseudo-random number generator that has no theoretical period?