Well, this question does lead deep into the gray area of what is a microprocessor and what has to be on chip to qualify as replacement, or do multi chip solutions also count, or when is a second chip an add on or an extension?
Not to mention the basic question what part of a mainframe is considered the CPU. Does the memory interface belong to the CPU or is it part of the memory (subsystem). Do IOCs count as part of the CPU or separate?
At least there is not doubt what a mainframe is, as there's nothing but /370(ish) (*1).
It maybe even does come down to the question, when is a microprocessor a microprocessor? There are not many chips that can be considered useful without supporting chips. A 8086 can only be used in very simple systems without 8282 and 8284 chips to decode its bus signals to be understood by 'normal' I/O or RAM/ROM.
Having said all that, there are many machines, that could be seen as microprocessor based /370 and alike:
Firstmost might be the IBM 5100. While not based on a single chip CPU (mind you, that was 1975), its discrete CPU did provide a mostly /370 instruction set.
Next there was the XT/370 in 1983, and its follow ups with the (main) implementation as one microprocessor. While the machine wasn't particularly fast (*2), it did fit with the lower end of what IBM offered as mainframes at that time.
But as you already mentioned, the true capability of a mainframe is its I/O performance. And that's something not even the later (1988) MCA implementation could do like the big ones.
During that time not only IBM, but also other mainframe manufacturers tried to sell microprocessor-based /370s. For example SIEMENS got its PC-2000, a single board add on for its Multi Bus based PC-MX2 Unix machines (NS32016 based). Fujitsu offered a similar setup. Again, due to the rather low I/O performance, they were only designated for development.
And then* there were also IBM (and other) mainframes using single chip CPUs, or better chip module, implementations starting in the late 1980s. First they were confined to the lower end, but nowadays next to all are based around such implementations. Strictly they are not single-chip but, then again, a Pentium II also is not.
Last but not least, today many lower to mid range systems are no longer based on 'real' implementations, but rather standard microprocessors running a /370 emulation. Fujitsu offers several mainframes based on Intel Xeon processors. Their actual machines can even be partitioned to run mainframe OS(es), Windows and Linux in parallel. This development is based on technologies they acquired when taking over Siemens-Nixdorf. Who again started it (codename Sunrise) when Siemens spun off their chip division (Infineon), so creating their chips couldn't be done in-house any more. At that time it was SPARC based, but now it's moved over to Intel. Still, the top end models are still based on custom implementations.
Before shunning emulation as 'not being the real thing' one should keep in mind, that almost every mainframe, beginning with the first /360s, was in part or complete delivered via a micro-programmed environment that changed many times below ISA level. The mainframe architecture never was about a certain real implementation, but it's an abstraction layer - or better a collection thereof.
*1 - Ok, there is another kind: Scientific processing with CDC and Cray as main players. But this kind never really had the pressure to put up with decades of ISA compatibility. Scientific applications, if they ever were used for more than a few years, had all source code available and just ported to the next generation anyway. There was never a real need to develop compatible microprocessor-based solutions - they got replaced right away by stock microprocessors.
*2 - 0.1 /370 MIPS may not sound fast, but at the same time a .9 MIPS machine with 1,25 MiB of memory was good to serve realtime data for 200+ individual terminal users.