Generative approaches powered by large language models (LLMs) have demonstrated emergent abilities in tasks that require complex reasoning abilities. Yet the generative nature still makes the generated content suffer from hallucinations, thus unsuitable for entity-centric tasks like entity linking (EL) requiring precise entity predictions over a large knowledge base. We present Instructed Generative Entity Linker (INSGENEL), the first approach that enables casual language models to perform entity linking over knowledge bases. Several methods to equip language models with EL capability were proposed in this work, including (i) a sequence-to-sequence training EL objective with instruction-tuning, (ii) a novel generative EL framework based on a light-weight potential mention retriever that frees the model from heavy and non-parallelizable decoding, achieving 4$\times$ speedup without compromise on linking metrics. INSGENEL outperforms previous generative alternatives with +6.8 F1 points gain on average, also with a huge advantage in training data efficiency and training compute consumption. In addition, our skillfully engineered in-context learning (ICL) framework for EL still lags behind INSGENEL significantly, reaffirming that the EL task remains a persistent hurdle for general LLMs.