Weather-sensitive resources are the main source of uncertainties in power systems. However, the unpredictable climate change further introduces ambiguity into the system, since the weather-sensitive resources would evolve with the climate and gradually exhibit a different probability distribution from the past in an uncertain manner. Lack of considering this climate-induced ambiguity in transmission network expansion planning (TNEP) may cause misunderstanding of future operational scenarios. Aiming at a higher security operation level under climate change yet less line investment, this paper proposes a climate-adaptive TNEP, which is essentially a robust TNEP equipped with a climate-adaptive uncertainty set (CUS). Determination of the CUS involves three steps. First, model future unknown distribution under climate change. Specifically, the climate-driven evolution in distributions is quantified by an evolutionary distance between historical and future true distributions, whose upper bound is derived from practical data, while the future unknown distribution is then modeled by a distance-based ambiguity set; Second, determine the CUS which has a minimal volume yet a desired confidence level in the face of the ambiguous future distribution. To that end, a parametric Wasserstein distance-based distributionally robust optimization (p-WDRO) is developed over the ambiguity set; Third, solve the p-WDRO by a data-clustering-incorporated reformulation. After the CUS is determined, the overall climate-adaptive TNEP is solved by a column-and-constraint-generation method with an inner multi-loop algorithm tailored for the CUS. Simulations are conducted on three test systems with practical data, which demonstrate that the climate-adaptive TNEP can improve operational security under climate change while reducing investment costs.