[EDIT] Since my various edits got quite long, maybe I can answer the question more directly and refer to previous versions for elaboration.
A key elementary result can be found in Feit's 1982 monograph The Representation Theory of Finite Groups (North-Holland Mathematical Library). His Chapters I to XII have no titles but are broken into numerous sections. Already in Chapter I he has a section 17, "Algebras over complete local domains", with an explicit result Corollary 17.13 which applies to a semisimple group algebra. (His conventions on $R, K, A$ are formulated at the beginning of the section, and apply to the group algebra of a finite group when $K$ has characteristic 0 and is the fraction field of a ring $R$ of $p$-adic integers, with finite residue field of characteristic $p>0$.)
This result shows that any simple $A$-module has an indecomposable reduction mod $p$. For $A=KG$, it just remains to notice that when such a reduction has two or more composition factors, there is also an $RG$-lattice admitting a direct sum as a module over the residue field. When $G= S_3$ and $p=3$, for example, the 2-dimensional standard module is irreducible in characteristic 0 but has two composition factors mod 3.
Since all representations in this case can be realized over $\mathbb{Z}$, the extension to $p$-adic integers in $\mathbb{Q}_p$ is harmless.
More generally, as I indicated originally, the Lie family $\mathrm{SL}_2(\mathbb{F}_p)$ provides many more examples if one considers Brauer characters along with ordinary irreducible characters.
P.S. I've looked further into the standard textbook literature, which is fairly limited in terms of giving detailed accounts of Brauer theory. In Serre's lectures (translated into English as Springer GTM 42), Exercise 15.2 is close to Ben's answer here, but similarly doesn't start with a simple module in characteristic 0. But the treatise by Curtis-Reiner Methods of Representation Theory in two volumes does have an exercise early in Chapter 2 (which introduces modular representations of finite groups in a modern style). This occurs on p. 416 at the end of $\S16$ (before the introduction of Brauer characters in $\S17$), as Exercise 3: "Find an example of a $p$-modular system $(K,R,k)$, a finite group $G$, and two full $RG$-lattices $M_1$ and $M_2$ in a simple $KG$-module $V$, such that $\overline{M}_1$ and $\overline{M}_2$ are not $kG$-isomorphic." (Here the bar indicates reduction modulo $p$ and the notation differs from that used by spin.)
[Alex Zalesskii has reminded me of the Feit result, which I had long ago bookmarked but then forgot about.]
