I wrote up some notes on this in 2004.  There have been some developments since then that I will indicate below.

Denote the smooth, proper morphism as follows, $$\pi:\mathcal{X}\to \text{Spec}\ R.$$  Denote by $\omega_\pi$ the relative dualizing sheaf.  Since $\pi$ is flat and proper, also the fiber product morphism, $$\mathbb{P}^n\times_{\text{Spec}\ R}\mathcal{X}\to \text{Spec}\ R,$$ is also flat and proper.  For a flat, proper morphism, Michael Artin proved representability of the relative Hilbert functor by an algebraic space that is locally finitely presented and separated over the base.  Thus, there exists a universal pair $$(\rho:I\to \text{Spec}\ R,\phi:I\times_{\text{Spec}\ R}\mathbb{P}^n \xrightarrow{\cong} I\times_{\text{Spec}\ R}\mathcal{X})$$ of a separated, locally finitely presented scheme $I$ over $\text{Spec}\ R$ and an isomorphism $\phi$ of $I$-schemes, and this is compatible with arbitrary base change of $\text{Spec}\ R$. 

<B>Question 1.</B>  Is $\rho$ surjective on points?

This can be checked after surjective base change of $\text{Spec}\ R$.  Thus, without loss of generality, assume that the generic fiber $X_K$ is $K$-isomorphic to $\mathbb{P}^n_K$.  Let $H\subset X_K$ be an effective Cartier divisor whose associated invertible sheaf generates the Picard group.  Since $\mathcal{X}$ is smooth over $\text{Spec}\ R$, also $\mathcal{X}$ is regular.  Thus the closure $\mathcal{H}$ of $H$ in $\mathcal{X}$ is a Cartier divisor that is flat over $\text{Spec}\ R$.  Denote by $\mathcal{L}$ the associated invertible sheaf (the dual of the ideal sheaf of $\mathcal{H}$).  The restriction of $\mathcal{L}$ to $X_K$ is an ample generator of $\text{Pic}(X_K)$.

<B>Lemma 1.</B> Let $\pi:\mathcal{X}\to \text{Spec}\ R$ be a proper, flat morphism and let $\mathcal{L}$ be an invertible sheaf on $\mathcal{X}$ whose restriction to $X_K$ is a generator of the Picard group.  If the closed fiber $X_k$ is integral, then every invertible sheaf on $\mathcal{X}$ is isomorphic to $\mathcal{L}^{\otimes m}$ for some integer $m$.

<B>Proof.</B>  Since $\pi$ is proper and flat with integral closed fiber, also $\mathcal{X}$ is integral.  Thus, every invertible sheaf on $\mathcal{X}$ is $R$-flat.  So the pushforward by $\pi$ is a finitely generated, torsion-free $R$-module, i.e., it is a finite, free $R$-module.

For every invertible sheaf $\mathcal{M}$ on $\mathcal{X}$, since $\mathcal{L}|_{X_K}$ generates $\text{Pic}(X_K)$, there exists an integer $m$ such that $\mathcal{M}|_{X_K}$ is isomorphic to $\mathcal{L}^{\otimes m}|_{X_K}$.  The pushforward $\pi_*\textit{Hom}_{\mathcal{O}_X}(\mathcal{M},\mathcal{L}^{\otimes m})$ is a finite, free $R$-module whose $K$-fiber equals $H^0(X_K,\mathcal{O}_{X_K})$.  Thus, there exists an element of this $R$-module whose $K$-germ is an invertible element of $H^0(X_K,\mathcal{O}_{X_K})$.  This element corresponds to a homomorphism of invertible sheaves, $$\alpha:\mathcal{M} \to \mathcal{L}^{\otimes m},$$ that is an isomorphism on $X_K$.  Thus, the support of the cokernel of $\alpha$ is an effective Cartier divisor on $\mathcal{X}$ that is disjoint from $X_K$.  

Since the closed fiber $X_k$ is integral, the support of the cokernel equals $r\underline{X}_k$ for some integer $r\geq 0$.  Thus $\mathcal{M}$ is isomorphic to $\mathcal{L}^{\otimes m}(-r\underline{X}_k)$ for some integer $r$.  Since $\mathcal{O}(-\underline{X}_k)$ is the pullback by $\pi$ of the maximal ideal of $R$, and since this is a free $R$-module, also $\mathcal{O}(-\underline{X}_k)$ is a free $\mathcal{O}_{\mathcal{X}}$-module.  Therefore, $\mathcal{M}$ is isomorphic to $\mathcal{L}^{\otimes m}$. <B>QED</B>

<B>Lemma 2.</B> The invertible sheaf $\mathcal{L}$ is $\pi$-ample.

<B>Proof.</B>  For every open affine $U \subset \mathcal{X}$ that intersects the closed fiber $X_k$, for every pair $p,q\in U$ of distinct closed points, there exists $f\in H^0(U,\mathcal{O}_{\mathcal{X}})$ that is zero on $p$ and that is nonzero on $q$.  The zero scheme $\text{Zero}(f)$ is a Cartier divisor in $U$.  The intersection with $U_k$ contains $p$, yet does not contain $q$.  The closure in $\mathcal{X}$ of $\text{Zero}(f)$ is a Cartier divisor $D$ that contains $p$, yet does not contain $q$.  By the previous lemma, the divisor $D$ is linearly equivalent to $m\mathcal{H}$ for some integer $m$.  Since $D$ is effective, its restriction to $X_K$ is effective.  Therefore $m$ is nonnegative.  Since $D$ is not the empty divisor, $m$ is positive.  Therefore, there exists a positive integer $m$ and a global section $s$ of $\mathcal{L}^{\otimes m}$ that vanishes at $p$ yet does not vanish at $q$.  Note that for every positive integer $d$, also $s^d$ is a global section of $\mathcal{L}^{\otimes md}$ that vanishes at $p$ yet does not vanish at $q$.

Combined with a Noetherian induction argument, it follows that for all sufficiently positive and divisible integers $m$, the invertible sheaf $\mathcal{L}^{\otimes m}$ is globally generated, and the complete linear system defines an injective morphism to projective space.  Since $\mathcal{X}$ is proper, this morphism is finite.  Hence, the invertible sheaf $\mathcal{L}$ is ample. <B>QED</B>

<B>Nota bene.</B> There is a vast generalization of Lemma 2 in the following article of Gounelas and Javanpeykar.

Frank Gounelas, Ariyan Javanpeykar<br>
Invariants of Fano varieties in families <br>
https://arxiv.org/abs/1703.0573

<B>I will complete the argument soon.</B>  The basic idea now is to use specializations of lines to prove that $\mathcal{H}$ is smooth over $\text{Spec}\ R$ (or else the multiplicity at a singular point would give an intersection number with a specialization of a line that is greater than one).  Then, by an induction hypothesis, the scheme $\mathcal{H}$ is a projective space bundle over $\text{Spec}\ R$.  Finally, fixing a $k$-point $q$ of $X_k$ that is not contained in $H_k$, the specializations in $X_k$ of lines that contain $q$ each intersect $H_k$ in a unique point.  From this, it follows that the blowing up of $X_k$ along $q$ is isomorphic to the projective bundle over $H_k$ of $\mathcal{O}_{H_k}\oplus \mathcal{L}|_{H_k}$.  Using the induction hypothesis, this equals the blowing up of $\mathbb{P}^n_k$ along a $k$-point.