Domination in graphs is a widely studied field, where many different definitions have been introduced in the last years to respond to different network requirements. This paper presents a new dominating parameter based on the well-known strong Roman domination model. Given a positive integer $p$, we call a $p$-strong Roman domination function ($p$-StRDF) in a graph $G$ to a function $f:V(G)\rightarrow \{0,1,2, \ldots , \left\lceil \frac{Δ+p}{p} \right\rceil \}$ having the property that if $f(v)=0$, then there is a vertex $u\in N(v)$ such that $f(u) \ge 1+ \left\lceil \frac{ |B_0\cap N(u)|}{p} \right\rceil $, where $B_0$ is the set of vertices with label $0$. The $p$-strong Roman domination number $γ_{StR}^p(G)$ is the minimum weight (sum of labels) of a $p$-StRDF on $G$. We study the NP-completeness of the \emph{$p$-StRD}-problem, we also provide general and tight upper and lower bounds depending on several classical invariants of the graph and, finally, we determine the exact values for some families of graphs.