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;
- $h, m \in \mathbb{N}$ 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 $0 \leq j \leq m$;
- $d_0 := [K \colon \mathbb{Q}_p]$.
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". The proof uses the same analytic techniques that go into the proof of the (much more famous) Krasner lemma.
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" 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";
- Krasner's paper "Remarques au sujet d'une note de J.-P. Serre...", in which he reproves the formula using his methods.