  Let $F$ be a number field, and $\pi$ a regular algebraic cuspidal automorphic
representation of $\mathrm{GL}_N(\mathbb{A}_F)$ of symplectic type. When $\pi$
is spherical at all primes $\mathfrak{p}|p$, we construct a $p$-adic
$L$-function attached to any regular non-critical spin $p$-refinement
$\tilde\pi$ of $\pi$ to $Q$-parahoric level, where $Q$ is the
$(n,n)$-parabolic. More precisely, we construct a distribution $L_p(\tilde\pi)$
on the Galois group $\mathrm{Gal}_p$ of the maximal abelian extension of $F$
unramified outside $p\infty$, and show that it interpolates all the standard
critical $L$-values of $\pi$ at $p$ (including, for example, cyclotomic and
anticyclotomic variation when $F$ is imaginary quadratic). We show that
$L_p(\tilde\pi)$ satisfies a natural growth condition; in particular, when
$\tilde\pi$ is ordinary, $L_p(\tilde\pi)$ is a (bounded) measure on
$\mathrm{Gal}_p$. As a corollary, when $\pi$ is unitary, has very regular
weight, and is $Q$-ordinary at all $\mathfrak{p}|p$, we deduce non-vanishing
$L(\pi\times(\chi\circ N_{F/\mathbb{Q}}),1/2) \neq 0$ of the twisted central
value for all but finitely many Dirichlet characters $\chi$ of $p$-power
conductor.
