I am sorry if I see this question only now, but since no one gave the following answer, it seems worth posting it.

There is a general formula for the number of extensions of degree $d$ of a $p$-adic field $K$, which is
$$
\# \{ L / K \mid [L \colon K] = d \} = \sigma(h) \cdot \sum_{j = 0}^m \frac{p^{m+j+1} - p^{2j}}{p - 1} \cdot (p^{\varepsilon_p(j) \cdot d \cdot d_0} - p^{\varepsilon_p(j - 1) \cdot d \cdot d_0})
$$
where:

 - $\sigma$ denotes the [sum of divisors function][1];
 - $h, m \in \mathbb{N}$ are the unique natural numbers such that $p \nmid h$ and $d = h \cdot p^m$;
 - $\varepsilon_p(j) := \sum_{k=1}^j p^{-k}$ if $j \geq 1$, $\varepsilon_p(0) := 0$ and $\varepsilon_p(-1) := -\infty$, *i.e.* $p^{\varepsilon_p(-1) \cdot d} = 0$. In particular, $p^{\varepsilon_p(j) \cdot d} \in \mathbb{N}$ if $-1 \leq j \leq m$;
 - $d_0 := [K \colon \mathbb{Q}_p]$.

This formula is due to Krasner, and has been proved in the paper "[Nombre des extensions d'un degré donné d'un corps $\mathfrak{p}$-adique][2]". The proof uses the same analytic techniques that go into the proof of the (much more famous) [Krasner lemma][3].

Using this formula one can find that the number of extensions of degree $p$ of a $p$-adic field $K$ should be $p + 1 + p^2 \cdot (p^{d_0} - 1)$. However, this is not in accordance with what happens for $K = \mathbb{Q}_p$, for which the number of extensions of degree $p$ is $p^2 + 1$ (see for example Proposition 2.3.1 in the paper "[A database of local fields][4]" by Jones and Roberts). I hope that there is no error in Krasner's formula, and I hope to edit this answer soon with a coherent explanation.

Finally, let me remark that this formula is related to Serre's "mass formula", which is valid in any characteristic. This formula says that a certain "count" of totally ramified extensions of a local, non-Archimedean field $K$ of degree $d$ is equal to $d$. More precisely,
$$
\sum_{L \in \Sigma_d} (\# \kappa)^{d - 1 - \mathrm{v}_K(\mathrm{disc}(L/K))} = d
$$
where $\Sigma_d$ denotes the set of totally ramified extensions of $K$ which have degree $d$, and $\kappa$ is the residue field of $K$. Observe that if $p \nmid d$ then the formula can be written simply as $\# \Sigma_d = d$.
Two useful references for this are:

 - Serre's original paper "[Une 'formule de masse' pour les extensions totalement ramifiées de degré donné d'un corps local][5]";  
 - Krasner's paper "[Remarques au sujet d'une note de J.-P. Serre...][6]", in which he reproves the formula using his methods.

  [1]: https://oeis.org/wiki/Sum_of_divisors_function
  [2]: http://alpha.math.uga.edu/~lorenz/KrasnerNombreExtensions.pdf
  [3]: https://en.wikipedia.org/wiki/Krasner%27s_lemma
  [4]: https://www.sciencedirect.com/science/article/pii/S0747717105001276
  [5]: https://gallica.bnf.fr/ark:/12148/bpt6k6234149b/f323.item
  [6]: https://gallica.bnf.fr/ark:/12148/bpt6k9813511g/f215.item