This question also has an answer here (in German):
Die Aussprache der Endung -ow
Travelling in the northeast of Germany, I have noticed that there are a lot of place names ending in -ow, but it is not pronounced phonetically, but rather like the English word “oh”, for example in Pankow.
Where does this pronunciation come from, and why is it pronounced differently to how it is spelt? I was wondering whether it comes from the extinct Prussian language?
Names with a silent w in the ending are mostly familiy names and place names of Slavic origin. You will find a lot of -ow names in Poland and -ov names in Czechia and Slovakia. Bit by bit the written final -ow found its way into German language and also became dominant over existing written -au names of Germanic origin and replaced them. At the same time the spoken final -au sound of Germanic origin became dominat over the pronounced final w consonant and so the w finally became a silent w in German language whereas it still is pronounced in Slavic languages. Check out this interesting Wikipedia article treating this subject.
Others have already linked to the Wikipedia article on “Namen auf -ow”. I’d like to add that your hypotheses were basically right, except regarding Old Prussian: Old Prussian was a Baltic, not a Slavic language, although both are closely related; and it was spoken further to the (North-)East.
In general, place names tend to be quite sticky; the people that live in a place may change, or their languages, or both, but the old name for the place continues to be used in most cases. It is just adapted in pronunciation and spelling over time.
A somewhat related example from the English-speaking world are place names ending in -cester, in England. They derive from the Latin word castra (fort) and have survived until today, in spite of the relatively short Roman presence there and of the language transitions since. Their pronunciation (or their spelling, depending on how you look at it) is regarded as unusual, as well: Worcester [ˈwʊstə].
The Berlin local district Pankow like other regional names ending with -ow was derived from a Slavic name of this region. Still today variants of the Sorbian language are spoken in eastern parts of Germany but Sorbian is not a German dialect. The ending "-w" is not pronounced in Sorbian.
These names are not derived from Old Prussian even though we do find many Sorbian loanwords in Old Prussian likely due to the geographic proximity.
