We show well-posedness of the $p$-Laplace evolution equation on $\mathbb{R}^d$ with square integrable random initial data for arbitrary $1<p<\infty$ and arbitrary space dimension $d\in\mathbb{N}$. The noise term on the right-hand side of the equation may be additive or multiplicative. Due to a lack of coercivity of the $p$-Laplace operator in the whole space, the possibility to apply well-known existence and uniqueness theorems in the classical functional setting is limited to certain values of $1<p<\infty$ and also depends on the space dimension $d$. We propose a framework of functional spaces which is independent of Sobolev space embeddings and space dimension. For additive noise, we show existence using a time discretization. Then, a fixed-point argument yields the result for multiplicative noise.