The BBDG decomposition theorem says that if $f\colon X \to Y$ is a projective morphism of finite type $\mathbf{C}$-schemes and $X$ is smooth of (pure) dimension $d$ then $\mathbf{R}f_*\mathbf{Q}_\ell[d]\cong \oplus {}^p\mathrm{R}^i f _*(\mathbf{Q}_\ell[d])$ is a direct sum of its perverse cohomology sheaves, and each ${}^p\mathrm{R}^i f _*(\mathbf{Q}_\ell[d])$ is semisimple.

Is it expected that the decomposition theorem holds over more general base schemes? Are there any positive results in this direction? (see the update below)

One issue with this possible generalization is that the semisimplicity part of the decomposition theorem holds only over an algebraically closed field. So, in general, one needs to impose some analogous condition on the base.

I think the statement is already non-trivial (and interesting) for $Y=\mathrm{Spec} \ \mathbf{C}[[T]]$ and $X$ a regular, projective $Y$-scheme.

It seems somewhat plausible that the decomposition theorem is correct for the example above since there is a version of the decomposition theorem for projective morphisms of analytic spaces due to Saito.

Update: As mentioned in the comments, ${}^p\mathrm{R}^i f _*(\mathbf{Q}_\ell[d])$ cannot be always semi-simple over $\mathrm{Spec} \ \mathbf{C}[[T]]$. But it still makes sense to ask if each ${}^p\mathrm{R}^i f _*(\mathbf{Q}_\ell[d])$ is a direct sum of IC-sheaves $\mathrm{IC}_{Z_i}(\mathcal{L}_i)$ for closed subscheme $Z_i\subset Y$ and local systems $\mathcal{L}_i$ on open subschemes $U_i \subset Z_i$.