I think that the four examples from Ovid given by blagae are not quite relevant to the question raised by the OP: all of them can be argued to show a clearly adjectival behavior and are not infrequent at all in Classical Latin. It is not correct to say, as blagae does, that these examples "occur sporadically". One can apply the typical tests to them in order to show their adjectival status (e.g. their compatibility with comparative and superlative degrees, etc.). These examples are not so different from typical ones like the following one from Caesar, where the adjectival behavior of florens is obvious. Examples like semper appetentes gloriae fuistis are indeed very frequent in Classical Latin and their adjectival nature is indisputable.
Ubii, quorum fuit civitas ampla atque florens (Caes. BG 4,3,3).
I understand that the OP was not really asking for this well-known fact, i.e., the adjectival usage of Latin present participles and their expected compatibility with esse. At least, Joonas's invented examples of copular constructions with verbal present participles like faciens est or dicens sum, which, in my opinion, are ungrammatical in Classical Latin (but see brianpck's comment above on the Latin of the Vulgate) seem to point to a different phenomenon, to a more interesting one or to a less obvious one. More relevant to this issue is the existence of examples like the following one from Cicero, where verbal behavior is indeed involved (cf. the presence of a direct object (aliquid), the participles here cannot be used in their comparative and superlative forms, etc.).
Videtis ut senectus sit operosa et semper agens aliquid et moliens (Cic. Sen. 26)
This example is commented on by Woodcock (1959: 79) in his A New Latin Syntax as follows:
"A participle can be predicated with part of esse without losing its verbal characteristics. It then forms with esse what is almost equivalent to a compound tense of the verb: Cic. de Sen. 26 videtis ut senectus sit operosa et semper agens aliquid et moliens 'You see how old age is laborious and always doing and contriving something.' Here agens and moliens, although they are parallel to the adjective operosa, describe senectus with reference to an activity in which it is engaged as well as to a quality it possesses. Therefore there is enough of the verbal notion present to justify the accusative object aliquid instead of an object genitive alicuius rei. This use of the present participle is not very common".
I think that Woodcock is wrong in claiming that in this case the participle "forms with esse what is almost equivalent to a compound tense of the verb". Given his statement (and his translation), this author seems to suggest that the verbal present participles agens and moliens are nominal predicates (on a par with the adjective operosa, which is indeed a nominal predicate, i.e. a "subject complement"). I disagree: these verbal present participles can (also) be analyzed as modifiers of the copular construction sit operosa.
So, to conclude, "Can esse be used with an adjectival present participle?" Yes, of course, but, as noted, I guess that this well-known fact is not the OP's question. Rather I understand that his question is: "Can esse be used with a verbal present participle (e.g. faciens est / dicens sum)? My (preliminary) answer is negative, at least wrt Classical Latin (but see brianpck's comment above on the Vulgate).