Let $k$ be an algebraically closed field of char $p>0$ and $X$ be a proper smooth variety over $k$ that is simply connected. Then we know $Pic(X)$ does not contain any $\ell$-torsion for $\ell \not= p$, so Picard variety $Pic^0$ is trivial and $Pic(X)$ is a finitely generated abelian group.
Do we have $Pic(X)[p]=0$ as well? What if $X$ is only assumed to be tamely simple connected?
What follows is more relevant to TKe's question in the comments than it is to the original question. Regarding the original question, I believe that Enriques surfaces in characteristic 2 provide the simplest and best-studied example of this phenomenon.
However, this phenomenon does happen in every characteristic. For every prime integer $p\geq 5$, let $k$ be an algebraically closed field of characteristic $p$, and let $\mu_{p,k}$ denote the $k$-group scheme $\text{Spec}(A)$ with multiplication map $m$ where $$A:= k[t]/\langle t^p - 1 \rangle, \ \ m^*(t) = t\otimes t\in A\otimes_k A.$$ Denote by $W$ the projective $k$-scheme $\text{Proj}(B)$ with an $\mathcal{O}(1)$-linearized action $s$ of $\mu_{p,k}$ where $$B=k[x_0,x_1,\dots,x_{p-1}], \ \ s^*(x_\ell) = t^\ell x_\ell.$$ As a $k$-scheme, $W$ is projective space of dimension $p-1$. The action $s$ is free on the complement of the $p$ coordinate points.
Denote by $q:W\to V$ the geometric quotient of the action $s$. This morphism is flat away from the images of the coordinate points. Denote by $Z\subset V$ the image of the set of the coordinate points, and denote by $U$ the open complement of $Z$ in $V$. As the target of a flat $k$-morphism from a smooth $k$-scheme, also $U$ is a smooth $k$-scheme. By SGA 2, for a sufficiently ample, sufficiently general hypersurface $T$ in $V$ that is disjoint from the finite scheme $Z$, the restriction map on fundamental groups, resp. Picard groups, from $V$ to $T$ is an isomorphism, and the same holds for the restriction map from $W$ to $q^{-1}(T)$.
Since $W$ is simply connected, also $q^{-1}(T)$ is simply connected. Thus, every finite, étale $T$-scheme, say $\nu:T'\to T$, admits a $T$-morphism from $q^{-1}(T)$. If $T'$ is integral, then the induced field extension $k(T)\to k(T')$ is a subextension of the field extension $k(T)\to k(q^{-1}(T))$. Since this field extension has degree $p$, it follows that $\nu$ is an isomorphism. Therefore $T$ is algebraically simply connected.
On the other hand, the standard character of $\mu_{p,k}$, i.e., inclusion of $\mu_{p,k}$ in $\mathbb{G}_{m,k}$, induces a $\mathbb{G}_{m,k}$-torsor on $T$. Geometrically, this is the quotient of $q^{-1}(T)\times_{\text{Spec}(k)}\mathbb{G}_{m,k}$ by the diagonal action of $\mu_{p,k}$. This $\mathbb{G}_{m,k}$-torsor is equivalent to an invertible sheaf. That invertible sheaf is $p$-torsion, since the standard character of $\mu_{p,k}$ is $p$-torsion.
