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This is not a definitive answer to the question. It's goal is simply to establish enough background and perspective on the question to explain why I think that the answer is likely to be No''.

The $p$-adic $L$-function does not interpolate infinitely many special values of the $L$-function of $E$. Rather, the $p$-adic $L$-function (and this is a feature of $p$-adic $L$-functions generally) interpolates critical values (in the sense of Deligne) of twisted $L$-functions of $E$. For an elliptic curve, the only critical value is $s = 1$; it is because one allows twisting that one ends up with a function, and not just a number.

A little more precisely: the $p$-adic $L$-function of an elliptic curve $E$ over $\mathbb Q$ interpolates the twisted central values, i.e. the values $L(f_{\chi},1),$ where $f$ is the weight two modular form attached to $E$ by the modularity theorem for elliptic curves, $\chi$ denotes a character of $p$-power conductor, and $f_{\chi}$ denotes the twist of $f$ by $\chi$. (There are some extra factors to do with interpolation and so on, which I will ignore here.)

More precisely again, supposing that $E$ has good ordinary reduction at $p$ (and perhaps some other technical conditions which I'll suppress) the $p$-adic $L$-function is an element of the completed group ring ${\mathbb Z}_p[[{\mathbb Z}_p^{\times}]]$, whose specialization under a finite order character $\chi:{\mathbb Z}_p^{\times} \rightarrow \overline{\mathbb Q}^{\times}$ is (essentially) the $p$-primary part of the algebraic part of the classical $L$-value $L(f_{\chi},1)$.

Now the complex $L$-function of $E$ determines (and is determined by) the isogeny class of $E$. The reason for this is that a consideration of the Euler product allows one to determine the traces of all Frobenius elements on the Tate modules of $E$, and hence by Faltings' theorem (the Tate conjecture) determine $E$ up to isogeny. On the other hand, the $p$-adic $L$-function does not have a corresponding Euler product, and so it is not clear (to me, at least) that it determines the $p$-adic Tate module of $E$ (and hence $E$, up to isogeny, or equivalently, the complex $L$-function of $E$).

One complication (at least at a psychological level) in thinking about this question is that in Iwasawa theory one often just thinks about the ideal in ${\mathbb Z}_p[[{\mathbb Z}_p^{\times}]]$ that is generated by the $p$-adic $L$-function, since it is this ideal which is (according to the main conjecture) supposed to relate to the Selmer group of $E$ over the $p$-adic cyclotomic tower. And this ideal surely won't determine $E$ up to isogeny; it is much too coarse a piece of information (e.g. it could just be the unit ideal in many cases, say if $E$ has no points over ${\mathbb Q}(\zeta_p)$ and the $p$-torsion part of Sha$(E)$ over this field is trivial (and maybe some condition on Tamagawa numbers).)

The $p$-adic $L$-function carries more information than this ideal, of course; it really does know about all those special values. But it's not clear to me exactly what to do with this information. So while I'm pretty sure that the answer to your question is No'', I'm not sure how to actually prove it either way.

1

This is not a definitive answer to the question. It's goal is simply to establish enough background and perspective on the question to explain why I think that the answer is likely to be No''.

The $p$-adic $L$-function does not interpolate infinitely many special values of the $L$-function of $E$. Rather, the $p$-adic $L$-function (and this is a feature of $p$-adic $L$-functions generally) interpolates critical values (in the sense of Deligne) of twisted $L$-functions of $E$. For an elliptic curve, the only critical value is $s = 1$; it is because one allows twisting that one ends up with a function, and not just a number.

A little more precisely: the $p$-adic $L$-function of an elliptic curve $E$ over $\mathbb Q$ interpolates the twisted central values, i.e. the values $L(f_{\chi},1),$ where $f$ is the weight two modular form attached to $E$ by the modularity theorem for elliptic curves, $\chi$ denotes a character of $p$-power conductor, and $f_{\chi}$ denotes the twist of $f$ by $\chi$. (There are some extra factors to do with interpolation and so on, which I will ignore here.)

More precisely again, supposing that $E$ has good ordinary reduction at $p$ (and perhaps some other technical conditions which I'll suppress) the $p$-adic $L$-function is an element of the completed group ring ${\mathbb Z}_p[[{\mathbb Z}_p^{\times}]]$, whose specialization under a finite order character $\chi:{\mathbb Z}_p^{\times} \rightarrow \overline{\mathbb Q}^{\times}$ is (essentially) the $p$-primary part of the algebraic part of the classical $L$-value $L(f_{\chi},1)$.

Now the complex $L$-function of $E$ determines (and is determined by) the isogeny class of $E$. The reason for this is that a consideration of the Euler product allows one to determine the traces of all Frobenius elements on the Tate modules of $E$, and hence by Faltings' theorem (the Tate conjecture) determine $E$ up to isogeny. On the other hand, the $p$-adic $L$-function does not have a corresponding Euler product, and so it is not clear (to me, at least) that it determines the $p$-adic Tate module of $E$ (and hence $E$, up to isogeny, or equivalently, the complex $L$-function of $E$).

One complication (at least at a psychological level) in thinking about this question is that in Iwasawa theory one often just thinks about the ideal in ${\mathbb Z}_p[[{\mathbb Z}_p^{\times}]]$ that is generated by the $p$-adic $L$-function, since it is this ideal which is (according to the main conjecture) supposed to relate to the Selmer group of $E$ over the $p$-adic cyclotomic tower. And this ideal surely won't determine $E$ up to isogeny; it is much too coarse a piece of information (e.g. it could just be the unit ideal in many cases, say if $E$ has no points over ${\mathbb Q}(\zeta_p)$ and the $p$-torsion part of Sha$(E)$ over this field is trivial (and maybe some condition on Tamagawa numbers).)

The $p$-adic $L$-function carries more information than this ideal, of course; it really does know about all those special values. But it's not clear to me exactly what to do with this information. So while I'm pretty sure that the answer to your question is No'', I'm not sure how to actually prove it either way.