Let me try to expand Petya comment in an explicit case, hoping that this can be useful.
Take $K >>0$ and consider the set of functions
$f_k:=e^{-kx^2}, \quad k=1,2, \ldots, K$.
Assume that we want to find an analytic function $f$ satisfying the requirement of the question and which is a linear combination of the $f_k$, namely
$f=\sum_{k=1}^K a_kf_k, \quad a_k \in \mathbb{R}$.
Then we must solve the following system of equations
$\sum_{k=1}^k A_ka_k =\sum_{i,j=1}^K A_{ij} a_ia_j$
$\sum_{k=1}^k B_ka_k =\sum_{i,j=1}^K A_{ij} a_ia_j$
$\sum_{k=1}^k B_ka_k =\sum_{i,j=1}^K B_{ij} a_ia_j$,
where
$A_k:=\int_{\mathbb{R}} f_k, \quad A_{ij}:=\int_{\mathbb{R}} f_i \cdot f_j$,
$B_k:=\sum_{n \in \mathbb{Z}} f_k(n), \quad B_{ij}:=\sum_{n \in \mathbb{Z}} f_i(n) \cdot f_j (n)$.
This system of equations geometrically describes the intersection of three non-empty quadrics hypersurfaces in $\mathbb{R}^K$, so one expects infinite solutions for $K$ big enough.