  The paper concerns the $q$-state Potts model (i.e., with spin values in
$\{1,\dots,q\}$) on a Cayley tree $\mathbb{T}^k$ of degree $k\geq 2$ (i.e.,
with $k+1$ edges emanating from each vertex) in an external (possibly random)
field. We construct the so-called splitting Gibbs measures (SGM) using
generalized boundary conditions on a sequence of expanding balls, subject to a
suitable compatibility criterion. Hence, the problem of existence/uniqueness of
SGM is reduced to solvability of the corresponding functional equation on the
tree. In particular, we introduce the notion of translation-invariant SGMs and
prove a novel criterion of translation invariance. Assuming a ferromagnetic
nearest-neighbour spin-spin interaction, we obtain various sufficient
conditions for uniqueness. For a model with constant external field, we provide
in-depth analysis of uniqueness vs.\ non-uniqueness in the subclass of
completely homogeneous SGMs by identifying the phase diagrams on the
"temperature--field" plane for different values of the parameters $q$ and $k$.
In a few particular cases (e.g., $q=2$ or $k=2$), the maximal number of
completely homogeneous SGMs in this model is shown to be $2^q-1$, and we make a
conjecture (supported by computer calculations) that this bound is valid for
all $q\ge 2$ and $k\ge2$.
