For this recently asked and answered question For primes sufficiently large, must digit products be zero? it was readily understood that the question was about base 10. Last year, James Maynard proved that there are infinitely many primes with no significant zeroes among their base 10 digits.
Primes like 113, 156841 and 12345678912379, which are of course quite small. For example, $1 \times 0 \times 9 = 0$, but $1 \times 1 \times 3 = 3$. Obviously it takes just one significant 0 to zero out the digit product.
But what about other bases? In binary, this question is Are there infinitely many Mersenne primes? which has yet to be settled.
In ternary, if we can prove that there are infinitely many primes of the form $3^n - 2$ (represented by several 2s with a single 1 as the least significant digit), that also proves there are infinitely many primes in that base with nonzero digit product.
And if we look at larger bases, like, say, base 1024, it seems unlikely that the set of primes with nonzero digit product could be finite, but that's just an intuition, far from a conclusive proof.
For which bases is this question yet to be settled? And are there bases for which we can be certain by mere virtue of the size of the base?