Multiplicative decomposition of arithmetic progressions in prime fields

We prove that there exists an absolute constant $c>0$ such that if an arithmetic progression $\mathcal{P}$ modulo a prime number $p$ does not contain zero and has the cardinality less than $cp$, then it cannot be represented as a product of two subsets of cardinality greater than $1$, unless $\mathcal{P}=-\mathcal{P}$ or $\mathcal{P}=\{-2r,r,4r\}$ for some residue $r$ modulo $p$.