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Sample-efficient offline reinforcement learning (RL) with linear function approximation has recently been studied extensively. Much of prior work has yielded the minimax-optimal bound of $\tilde{\mathcal{O}}(\frac{1}{\sqrt{K}})$, with $K$ being the number of episodes in the offline data. In this work, we seek to understand instance-dependent bounds for offline RL with function approximation. We present an algorithm called Bootstrapped and Constrained Pessimistic Value Iteration (BCP-VI), which leverages data bootstrapping and constrained optimization on top of pessimism. We show that under a partial data coverage assumption, that of \emph{concentrability} with respect to an optimal policy, the proposed algorithm yields a fast rate of $\tilde{\mathcal{O}}(\frac{1}{K})$ for offline RL when there is a positive gap in the optimal Q-value functions, even when the offline data were adaptively collected. Moreover, when the linear features of the optimal actions in the states reachable by an optimal policy span those reachable by the behavior policy and the optimal actions are unique, offline RL achieves absolute zero sub-optimality error when $K$ exceeds a (finite) instance-dependent threshold. To the best of our knowledge, these are the first $\tilde{\mathcal{O}}(\frac{1}{K})$ bound and absolute zero sub-optimality bound respectively for offline RL with linear function approximation from adaptive data with partial coverage. We also provide instance-agnostic and instance-dependent information-theoretical lower bounds to complement our upper bounds.