Let $B^{\alpha}_p:=B^{\alpha}_{p,\infty}$ be the Besov space of regularity $\alpha<0$ and integrability $p\ge1$. Recall that a distribution $f$ from the dual Schwarz space is in $B^{\alpha}_p$ if and only if $$ \sup_{t\in(0,1]} t^{-\alpha/2}\|P_t f\|_{L_p(\mathbb{R})}<\infty, $$ where $P_t$ is the heat kernel. It is well-known that following Besov embedding holds for any $\alpha\in\mathbb{R}$, $p\ge1$: $$ B^{\alpha+1/p}_p\subset B^{\alpha}_\infty $$ and thus $$ \bigcup_{p\ge1} B^{\alpha+1/p}_p\subset B^{\alpha}_\infty. $$
My question is whether this embedding is strict. Namely, is it possible (for some $\alpha\in\mathbb{R}$) to find a function $f$ which belongs to $B^{\alpha}_\infty$ but not to $B^{\alpha+1/p}_p$ for any finite $p$?