Let α([0,1]^p) denote the intersection local time of p independent d-dimensional Brownian motions running up to the time 1. Under the conditions p(d−2)<d and d≥2, we prove $$\lim_{t\to\infty}t^{-1}\log \mathbb{P}\bigl\{\alpha([0,1]^{p})\ge t^{(d(p-1))/2}\bigr\}=-\gamma_{\alpha}(d,p)$$ with the right-hand side being identified in terms of the the best constant of the Gagliardo–Nirenberg inequality. Within the scale of moderate deviations, we also establish the precise tail asymptotics for the intersection local time $$I_n=\#\{(k_1,…,k_p)∈[1,n]^p ; S_1(k_1)=⋯=S_p(k_p)\}$$ run by the independent, symmetric, ℤ^d-valued random walks S₁(n), …,S_p(n). Our results apply to the law of the iterated logarithm. Our approach is based on Feynman–Kac type large deviation, time exponentiation, moment computation and some technologies along the lines of probability in Banach space. As an interesting coproduct, we obtain the inequality $$\bigl({\mathbb{E}}I_{n_{1}+\cdots +n_{a}}^{m}\bigr)^{1/p}\le \sum_{\mathop{k_{1}+\cdots +k_{a}=m}\limits_{k_{1},\ldots,k_{a}\ge 0}}{\frac{m!}{k_{1}!\cdots k_{a}!}}\bigl({\mathbb{E}}I_{n_{1}}^{k_{1}}\bigr)^{1/p}\cdots \bigl({\mathbb{E}}I_{n_{a}}^{k_{a}}\bigr)^{1/p}$$ in the case of random walks.