First of all, there is no single accepted definition of "neuromorphic" [1]. Still, as a point in favour of the "neuromorphic systems are analogue" crowd: the seminal paper by Carver Mead that (to my knowledge) coined the term "neuromorphics" specifically talks about analogue neuromorphic systems [2].
Right now, there are some research "analogue" (or, more precisely "mixed signal") neuromorphic systems being developed [3, 4]. It is correct however that there are no commercially available analogue systems that I am aware of.
Unfortunately, the same can be said for digital neuromorphics as well (Intel Loihi is perhaps the closest to a commercial product, and yes, this is an asynchronous digital neuromorphic system).
