Sampling the function to get a time series makes a vector-valued random variable out of a stochastic process, and numerical differentiation on that random vector is still deterministic.

This is not correct. Given the vector of all values of A sampled at intervals dt, the derivative of that vector -- that is, the time series for B -- is not determined by the vector itself, only by the complete trajectory of A. The longer dt is, the less the vector tells you about B.