This answer is primarily based on Ignace Feuerlicht, "Omissions and Contradictions in Kafka's Trial", The German Quarterly 1967, 40(3), pp. 339-350 - available here if you have Jstor access. All quotes below are from this article unless otherwise stated.
As mentioned in the OP, there is little explicit information in the text indicating a precise location. However, there are various hints which we can piece together using close reading.
First off, let's note something which may seem to be relevant evidence but actually isn't, and which is only apparent from the original German text rather than translations:
Words, such as Elektrische (47), bloßfüßig
(48, 173), Verkühlung (223, 238), and the use of sein in the perfect
of stehen (90, 224) and sitzen (160) point to Southern Germany
and Austria, but are indicative of Kafka's rather than K.'s country
As noted in the OP, we know that the setting isn't Italy or (more importantly) Germany. But so far this is all negative evidence. What positive indications do we have about where it is rather than where it isn't?
General cultural indications:
K. lives in a country which is at peace and where law and
order prevail (12).
The phrase mit keinem Heller bezahlen (138) and the
fact that Captain Lanz greets Miss Montag by bowing to her and
respectfully kissing her hand (100) suggest more strongly the old
Now let's start to get more specific. The following piece of evidence cuts down the possibilities a LOT:
K. [...] lives and works in a "capital" (112)
where German is spoken, judging from the first names and family
names (Elsa, Karl, Leni, Hasterer, Biirstner, Rabensteiner) and
from the name of the one and only street mentioned (JuliusstraBe).
Many or most of the inhabitants of that capital are Catholics since
the Cathedral is Catholic. This excludes Bern, the Protestant capital
Finally, most specifically of all, here's a quote from chapter 10 of The Trial itself (using the David Wyllie translation freely available here):
The moonlight glittered and quivered in the water, which divided itself around a small island covered in a densely-piled mass of foliage and trees and bushes. Beneath them, now invisible, there were gravel paths with comfortable benches where K. had stretched himself out on many summer’s days.
An island on a river with a park excludes Vienna, Linz, Graz, Salzburg, Klagenfurt, and Innsbruck, but it could fit Prague - namely Kampa Park on an island in the river Vltava.
Prague also fits the other clues mentioned above: it was the capital of the Kingdom of Bohemia within the Austro-Hungarian empire, a predominantly Catholic region where German was an official language. It also happens to be the city where Kafka was born and lived for most of his life, so it makes sense for him to use it as a setting for his book.
People, very familiar with Prague, recognize
indeed certain sections of the city, the Kampa Island, the Charles
Bridge, and especially the Cathedral Square and the Cathedral with
its silver statue of a saint.2
Finally, the Prague theory is also supported by Heinz Politzer's book Franz Kafka: Parable and Paradox, which says:
German Rabensteiner, the Czech Kullich, and the Jew Kaminer
represent the three nationalities of Prague, which is also the city of K.
TL;DR: it's set in Prague.
1 Page numbers are from Franz Kafka, *Der Prozeß*
(Berlin, 1951). H stands for *Hochzeitsvorbereitungen auf dem
Lande* (New York, 1953) and T for *Tagebucher* (New York, 1948).
2 Cited to Pavel Eisner, "Franz Kafkas Prozess und Prag", German Life and Letters 1961, 14: pp. 16–25 (in German; available here).