That conjecture is false without further hypotheses, e.g., flatness of $p$.  Consider the case where $A$ equals $k[s]$, where $X$ is the effective Cartier divisor of $\mathbb{P}^2_A = \text{Proj}\ A[T,U,V]$ with defining equation $sT$, and where $\mathcal{E}$ equals $\mathcal{F}$ equals the structure sheaf of the closed subscheme with defining ideal $\langle T,U \rangle$.  For $i=0$, the formation of $\text{Hom}_{\mathcal{O}_X}(\mathcal{E},\mathcal{F})$ is compatible with arbitrary base change.

However, the functor $\text{Ext}^1_{p_B}(\mathcal{E},\mathcal{F})$ is not representable in the way you describe.  For the base change to $A[1/s]$, it is representable by a locally free sheaf $Q_{1/s}$ of rank $1$.  Also, for the base change to $A/sA$, it is representable; but now the locally free sheaf is just the zero sheaf.  There is no finitely generated $A$-module $Q$ such that $Q\otimes_A A[1/s]$ is free of rank $1$ yet $Q/sQ$ is the zero sheaf.