Do hand-based hash functions / MACs exist? Is there research on a hash function or a MAC that can be executed without computers?

I know there's a lot of history on hand-based ciphers and the OTP can easily be excecuted by hand. (By using numbers / letters) But modern cryptography requires authentication as well as encryption.

Now I'm asking myself, is there a (simple) well-studied hash function or a similar MAC that can be executed by hand and operates on letters rather than bits?

Reformulation: Is there a "classical" analogon for hash functions to the classical ciphers.

Note: Resistance against computer-based attacks isn't a neccessary requirement (just like classical ciphers do not resist such attacks).
It depends on the time you want to spend. But most likely, there is nothing with reasonable efficiency. For arithmetic operations, humans are really bad compared to computers, and the difference is at least a factor of $10.000.000$ (very very rough guess, probably even 1+ additional zeros there).

So, since you have to assume that the attacker has access to a computer and will use it, you will have to use a modern scheme and consider classical ones broken.

Anyway, for security in a modern sense, you need a modern scheme, and those require modern tools like computers to achieve any reasonable performance. Of course you can calculate SHA256 by hand... but it will probably take you hours to go through all rounds for a single hash value.

Considering your remark about letters instead of bits: Mostly that is just interpretation, but there is also format-preserving encryption (there are quite a few questions on that on cryptoSE), which operates on arbitrary sets of symbols. If you want to transfer this into a hash fucntion, see this question.

Hash functions in the classic crypto-era?

At the time when classic ciphers were created, cryptography was (as far as I can tell) almost only looking at encryption methods. The idea of hash functions - or its ideal counter part the one-way function - might have been known, but not its range of application. E.g. integrity was not much of a concern: Someone without the key couldn't possibly modify a message in a meaningful way. Other applications of hash functions might seem trivial these days, but as always: developing something new and understanding something in hindsight is entirely different.

I doubt there is a genuine classical hash function.