Where is the root morpheme in Modern English ambassador, embassy?
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":
1570–80; variant of ambassy < Middle French ambassee, Old French ambasce, ambaxee
"-ador" is not an English suffix:
1325–75; Middle English am-, embass ( i ) adour, imbassadore < Anglo-French ambassateur, ambassaduer < Italian ambassatore, dialectal Italian ambassadore, equivalent to ambass- ( see embassy) -atore, -adore < Latin -ātōrem accusative of -ātor -ator(Emphasis added)
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.
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:
Adopted from Fr. ambassade, 15th c. ambaxade, ad. OSp. ambaxada (mod. em-), cogn. w. Pr. ambaissada, It. ambasciata, OFr. ambassée, (superseded by this form in -ade: see ambassy):—L. *ambactiāta (found in med.L. as ambaxiāta, -asciāta, -assiāta, -asiāta), ppl. derivative of *ambactiāre to go on a mission, f. ambactia, ambaxia (in Salic and Burgundian Laws) ‘charge, office, employment,’ n. of office f. ambactus a servant (? vassal, retainer). The OFr. form ambassée was also adopted in Eng. as ambassy, embassy; as was also the med.L. as ambassiate, etc., the forms of which appear to have been quite mixed up with those of the present word, leading to the pronunciation in 5–6 am'bassiade, am'bassade, and the spellings in -ad, -ed, -et. But Shakespeare and subseq. writers have amba'ssade or 'ambassade.
The origin and meaning of ambactus have given rise to much discussion. According to Festus ‘Ambactus apud Ennium lingua Gallica servus appellatur’; and Caesar (B.G. vi. 15) applies it to the vassals or retainers of a Gallic chief. Hence Zeuss and Glück identify it with Welsh amaeth, ammaeth, (for *ambaeth) ‘husbandman, tiller of the ground,’ perh. orig. ‘tenant, retainer,’ or even ‘goer about, footman.’ Grimm finds the origin in OHG. ambaht, Goth. andbahts servant, retainer, OE. ambeht, ON. ambótt (cf. amboht), variously explained as f. and against, towards + bak back, or *bah to do, or *baht = Skr. bhakta devoted, and assumed to have been adopted in Gallic, or erroneously taken as Gallic by Festus. But the majority of etymologists consider the Teut. word to be an adaptation or refashioning of the Lat. or original Celtic. For the latter, Mahn (Etym. Unt. 145) has also proposed ambi(amb-, amm-, am-) about + Breton aketuz, akeduz ‘busy,’ hence ‘one employed about (his lord).’