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 <B>Severi-Brauer variety</B>, i.e., a "twist" of projective space $\mathbb{P}^{n-1}$ (the simplest proper scheme).  All such twists arise from central simple algebras over the field.  If the central simple algebra is a <b>cyclic algebra</b> (e.g., every Brauer class over a local or global field is represented by a cyclic algebra by class field theory), then there is always a <B>unique</B> $\text{Aut}(Y)$-orbit of <i>nodal</i> curves $X_0$ of arithmetic genus $1$ in $Y$ that are <i>elliptic normal curves</i> 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 <b>smooth</b> elliptic normal curves $X$ in $Y$ obtained as deformations of $X_0$.    

Recall the definition of <b>cyclic algebras,</b> as in Roquette's beautiful history of class field theory.

MR2222818 (2006m:11160) <br>
Roquette, Peter <br>
The Brauer-Hasse-Noether theorem in historical perspective. <br> 
Schriften der Mathematisch-Naturwissenschaftlichen Klasse der Heidelberger Akademie der Wissenschaften, 15. <br>
Springer-Verlag, Berlin, 2005. vi+92 pp. ISBN: 3-540-23005-X <br>
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 <I>cyclic algebra</I> 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 <b>Severi-Brauer variety</b> 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 $K$-subscheme of dimension $0$ and length $n.$  

In particular, the <b>index</b> of $Y$, i.e., the least positive length of a $0$-dimensional closed $K$-subscheme of $Y$, equals $n.$  So for cyclic algebras of period $n,$ i.e., the order of the corresponding Brauer class in the Brauer group of the field, the <i>index equals the period</i>, irrespective of the field $K.$   The Merkurjev-Suslin Theorem, and the refinements by Merkurjev, imply that for every prime integer $n$, the $n$-torsion subgroup of the Brauer group is generated (as a group) by the classes of cyclic algebras as above.  Nonetheless, a typical element of this group is not represented by a cyclic algebra, and there are many examples of central simple algebras whose index is much larger than the period (the period divides the index, and they have the same list of prime factors).  It is an open problem to relate the symbol length of an algebra (i.e., the fewest number of cyclic $n$-algebras necessary to generate a specified $n$-torsion Brauer class) and the index, e.g., it is an open problem whether every Brauer class over the function field of a complex surface is represented by a cyclic algebra (for these fields, the index does equal the period by de Jong's Period-Index Theorem).  Finally, for a local or global field $K$, every $n$-torsion Brauer class is represented by a cyclic $n$-algebra.  This is a consequence of the Brauer-Hasse-Noether(-Albert) Theorem and Hasse's Structure Theorem from class field theory.

For the Severi-Brauer variety of a cyclic algebra, there is a unique $\text{Aut}(Y)$-orbit of the paramter space of <i>nodal, elliptic normal curves</i> with linear (geometric) components, and this orbit has a $K$-point parameterizing such a curve $X_0.$  Here is the construction.  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 <b>nodal</b>: 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.

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". 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 $Y$.