There are already two great answers, but I want to post an answer that works over all local fields, such as $\mathbb{Q}_p$, based on an alternative philosophy. Instead of starting with an elliptic curve $(E,0)$ and studying torsors $X$ for that curve that have large index, first we start with a "simpler" ambient scheme $Y$ that manifestly has large index, and then we try to find a genus $1$ curve $X$ in that scheme. The simpler scheme is a twist of projective space $\mathbb{P}^{n-1}$ (the simplest proper scheme in my opinion). All such twists arise from central simple algebras over the field. If the central simple algebra is a "cyclic algebra" (and that includes all central simple algebras over local and global fields by class field theory), then there is always a unique $\text{Aut}(Y)$-orbit of nodal curves $X_0$ of arithmetic genus $1$ in $Y$ that are elliptic normal curves with linear components. For instance, in $\mathbb{P}^2$, the curves $X_0$ are unions of three non-concurrent lines. The deformation theory of $X_0$ in $Y$ is unobstructed. Thus, if the field is "ample" or "large" in the sense of Florian Pop (and this includes all local fields), then there exist smooth elliptic normal curves $X$ in $Y$ obtained as deformations of $X_0$.
Recall the definition of cyclic algebras, as in Roquette's beautiful history of class field theory.
MR2222818 (2006m:11160)
Roquette, Peter
The Brauer-Hasse-Noether theorem in historical perspective.
Schriften der Mathematisch-Naturwissenschaftlichen Klasse der Heidelberger Akademie der Wissenschaften, 15.
Springer-Verlag, Berlin, 2005. vi+92 pp. ISBN: 3-540-23005-X
https://www.mathi.uni-heidelberg.de/~roquette/brhano.pdf
Let $K$ be a field, and let $n$ be an integer prime to the characteristic. Let $L/K$ be a finite field extension of degree $n$ that is Galois with cyclic Galois group $\langle \sigma \rangle.$ Let $a\in K^\times$ be an element. The cyclic algebra over $K$ associated to $L$ and $a$ is the $L$-vector space of dimension $n$, $$A(L/K,\sigma,a) := L\cdot 1 \oplus L\cdot u \oplus \dots \oplus L\cdot u^{n-1}, $$ with the unique $K$-central algebra structure determined by the relations, $$1\cdot x = x = x\cdot 1, \ \ u^n = a\cdot 1, \ \ u\cdot b = \sigma(b) \cdot u,$$ for every $x\in A(L/K,\sigma,a)$ and for every $b\in L.$ This is a central simple $K$-algebra.
The Severi-Brauer variety of $A(L/K,\sigma,a)$ is the smooth, projective $K$-scheme $Y$ that represents the functor on $K$-schemes $T$ associating to $T$ the set of all left ideals in $A(L/K,\sigma,a)\otimes_K \mathcal{O}_T$ that are locally direct summands of (free) rank $n.$ There is an evident inclusion of $L$ in $A(L/K,\sigma,1)$ as $L\cdot 1.$ That inclusion induces a left ideal in $A(L/K,\sigma,a)\otimes_K L$ giving an $L$-point of $Y.$ The Galois orbit of this $L$-point, $\Gamma \subset Y$, is a smooth closed subscheme of dimension $0$ and length $n$ that is defined over $K.$ The automorphism $\sigma$ restricts to an automorphism of $\Gamma.$ There is a unique minimal closed subscheme $X_0\subset Y$ that contains $\Gamma,$ whose geometric irreducible components are lines, and such that for every geometric point $p:\text{Spec}\kappa \to \Gamma$, there is a $\kappa$-irreducible component of $X_0$ that contains both $p$ and $\sigma(p).$ Concretely, after base change to $L$, the union of the $n$ lines $\Lambda_r=\text{span}(\sigma^r(p),\sigma^{r-1}(p))$, $r=0,\dots,n-1,$ is Galois-invariant, hence equals the base change of a $K$-curve $X_0\subset Y$.
The $K$-curve $X_0$ is geometrically connected and geometrically reduced. The curve $X_0$ is nodal: the point $\sigma^r(p)$ is contained in two irreducible components $\Lambda_r$ and $\Lambda_{r+1}.$ The arithmetic genus of $X_0$ equals $1$. Geometrically, $X_0$ is an elliptic normal curve in $\mathbb{P}^{n-1},$ i.e., it is linearly nondegenerate and linearly normal (necessarily of degree $n$). In fact, any two such curves in $\mathbb{P}^{n-1}$ are conjugate under the group $\text{Aut}(\mathbb{P}^{n-1})$ of projective linear transformations. Thus, there is a unique $\text{Aut}(X)$-orbit of such curves $X_0$ in $Y.$
Finally, an obstruction group for infinitesimal deformations of a curve $X$ in $Y$ with ideal sheaf $\mathcal{I}$ is $$O_{X,Y}=\text{Ext}^1_{\mathcal{O}_X}(\mathcal{I}/\mathcal{I}^2,\mathcal{O}_X).$$ This is compatible with base change from $K$ to $L$, where $Y_0$ equals a union of $n$ lines. Since the normal bundle of $\Lambda_r$ equals $\mathcal{O}(1)^{\oplus (n-2)}$, and since even after twisting down by $\sigma^r(p)$ and $\sigma^{r-1}(p)$, the twisted sheaf $\mathcal{O}(-1)$ on the line has vanishing $h^1$, it follows that the obstruction group is the zero group, and infinitesimal deformations smooth all nodes. Thus, the Hilbert $K$-scheme parameterizing elliptic normal curves in $Y$ is smooth at the point parameterizing $X_0,$ and the unique irreducible component of the Hilbert scheme containing this point has a dense open subscheme $U$ parameterizing smooth elliptic, normal curves.
Finally, if the field $K$ is "ample" or "big" in the sense of Florian Pop, then there are $K$-points of $U$. In particular, by Hensel's Lemma, the fraction field of every Henselian DVR is "big". Finally, for every local field $K$, by the Brauer-Hasse-Noether theorem (also proved by Albert) and by Hasse's structure theorem, every central simple algebra over a local or global field is cyclic. In particular, for a local field $K$ such as $\mathbb{Q}_p$ or $\mathbb{F}_p((t)),$ every period-$n$ element in the Brauer group $\text{Br}(K)\cong \mathbb{Q}/\mathbb{Z}$ is represented by a cyclic algebra as above, and there exist $K$-points of $U$ parameterizing smooth elliptic normal curves $X$ in $Y$.