Is there a known example of a non-smooth irreducible component of the rigid generic fibre of a Hida family?

Let me explain some of the context around this question (but I'm not going to explain Hida theory -- he wrote a book on it if anyone is interested!). Let $\Lambda$ be the ring $\mathbf{Z}_p[[T]]$ as is usual in Hida theory. Given a level $N$ and a prime $p$ Hida constructs a semi-local $\Lambda$-algebra $\mathbf{T}$ interpolating ordinary Hecke algebras of level $N$ and all weights (in a congruence class mod $p-1$ (or mod $2$ if $p=2$)), and this algebra is typically finite and free as a $\Lambda$-module. In particular it gives rise to a finite flat cover of the rigid analytic generic fibre of $\Lambda$, which is a $p$-adic open disc. Let me call this rigid space the "Hida generic fibre".

One might ask what Hida generic fibres can look like. Certainly they have finitely many irreducible components. Examples are known where these components map down in a ramified way down to weight space; examples are known where two irreducible components cross, giving rise to singularities in the generic fibre. Components crossing can happen for several reasons, for example a component which is generically new at some auxiliary prime $\ell$ dividing $N$ once can meet a component which is generically old at $\ell$, a CM family can meet a non-CM family, or two CM families corresponding to different imaginary uadratic fields can meet (in a weight 1 point for example). All of these give examples of phenomena where the map from the Hida rigid space down to weight space is not etale (and although my memory is getting rusty these phenomena might be all the examples I know of how the map down to weight space can fail to be etale). But in all these cases it could happen that every component of the Hida generic fibre is smooth. I don't know any examples where a component is not smooth. Does anyone?