If there were no such a word as embassy, I would consider ambassad as a root and -or as an agent derivational suffix here. But embassy makes me puzzled. If we accept that segmentation shold be done like EMBASS-Y, how can we define -ador in ambassador then? as a derivational suffix?
"Embassy" is actually a variant of the word "ambassy":
"-ador" is not an English suffix:
Note that it is not an English suffix, but Latin. we just derived the word from the Latin word, including all the affixes.
So, to answer your question, ambassador in English is a word by itself. You can't break it down to root and suffixes (no suffix as -ador in English) in English, only if you went back to its Latin roots, then can you derive the root word. So, in English, ambassador is a root by itself, for words like "ambassadorial", "ambassadorship", etc.
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The point is that we borrowed both ambassador and embassy whole from other languages: neither word was assembled in English. For that reason, it is better to say that the -or in ambassador is the Latin/Italian suffix, not the English one. But that is immaterial, since both suffixes come from the same Proto-Indo-European root and have similar meanings.
Both ambassador and embassy come from the Latin word ambactus, "servant". Going further back in time, the etymology becomes murky. According to the Oxford English Dictionary (2nd edition), the Latin word may have been derived from Celtic, but that is uncertain. Alternatively, it may come from the Proto-Germanic precursor of modern Dutch/German ambacht. But most etymologists hold the view that the Germanic word must have been derived from either the Celtic or the Latin word.
The OED on ambassade: