We consider vertex partitions of the binomial random graph $G_{n,p}$. For $np\to\infty$, we observe the following phenomenon: for any partition into asymptotically fewer than $\chi(G_{n,p})$ parts, i.e. $o(np/\log np)$ parts, there must be one part whose induced subgraph has a connected component of order at least roughly the average part size. Stated another way, we consider the $t$-component chromatic number, the smallest number of colours needed in a colouring of the vertices such that each colour class induces a subgraph of maximum component order at most $t$. As long as $np \to \infty$, there is a threshold around $t = \Theta(\log_b np)$ (where $b = 1/(1-p)$), such that if $t$ is smaller then the $t$-component chromatic number is about as large as the chromatic number, whereas if $t$ is greater then it is close to the trivial upper bound $n/t$. For $0 < p <1$ fixed, we obtain more precise information. In particular, we find something more subtle happens at the threshold $t = \Theta(\log n)$, and we determine the asymptotic first-order behaviour. Moreover, we consider the $t$-component stability number, the maximum order of a vertex subset that induces a subgraph with maximum component order at most $t$, and show that it is concentrated in a constant length interval about an explicitly given formula, so long as $t = O(\log \log n)$.