Let $R$ be a DVR of mixed characteristic, with algebraically closed residue field of characteristic $p$ and fraction field $K$ (I'm mainly interested in the case $R=W(k)$). Let $Y\longrightarrow \operatorname{Spec} R$ be a smooth projective morphism of relative dimension 2. Let $f:Y\longrightarrow \mathbb{P}_R^1$ be a morphism admitting a smooth section, such that $f_K: Y_K\longrightarrow\mathbb{P}_K^1$ and $f_k: Y_k\longrightarrow\mathbb{P}_k^1$ are elliptic surfaces of Kodaira dimension one, whose only singular fibers are of type $I_m$ ($m$ is allowed to vary across the singular fibers). Is there an example of $Y$ as above so that, furthermore, the specialization map $$Pic^0(Y_K/\mathbb{P}^1_K)[p]\longrightarrow Pic^0(Y_k/\mathbb{P}^1_k)[p]$$ is not injective?