As many researchers in the area, I have been living with the false belief that a promise-BPTime hierarchy is easy to prove by diagonalization. But as pointed out by Songhua He and briefly explained below, this is not the case. Nevertheless, Songhua He proves such a time hierarchy by using delayed diagnonalization. Needless to say, this is a great service to our community.
The flawed argument proceeds by considering an efficient enumeration of probabilistic machines (with infinitely many repeats of each machine) and an efficient enumeration of strings such that the $n^\xth$ string, denoted $x_n$, has length $n$. We let $x_n\in\YES$ (resp., $x_n\in\NO$) if, on input $x_n$, with probability at least~$2/3$, the $n^\th$ machine halts within $t(n)$ steps with output~0 (resp., with output~1). It follows that $\Pi=(\YES,\NO)$ is in $\prBPTIME(\tildeO(t))\setminus\prBPTIME(t)$.
Indeed, $\Pi$ is in $\prBPTIME(\tildeO(t))$, but the claim that it is not in $\prBPTIME(t)$ is not supported. Indeed, if the $n$th machine, denoted $M_n$, decides some set in $\BPTIME(t)$, then we failed it on $x_n$. However, in general, (the time-modified) machine $M_n$ may decide a promise problem, and the input $x_n$ may violate the promise. In the latter case, $M_n$ was not adequately failed (i.e., it has to be failed on an input that satisfies the promise). There is no easy fix, since we cannot tell (efficiently) whether or not an input violates the promise of $M_n$.
We prove a time hierarchy theorem for the promise-BPTIME. This is considered to be a folklore problem and was thought to follow from the existence of complete problems or through direct diagonalization. We observe that neither argument carries through in some immediate way in the promise version. However, the hierarchy theorem can be proved by the standard delayed diagonalization for the nondeterministic time hierarchy theorem [Coo72, SFM78, Zak83], or, as it was observed by Rahul Santhanam, from the established BPTIME hierarchies with advice [Bar02, FS04, GST11, FST05, Per05, vMP07, San25].
See ECCC TR25-004.