Cooperative perception (CP) offers significant potential to overcome the limitations of single-vehicle sensing by enabling information sharing among connected vehicles (CVs). However, existing generic CP approaches need to transmit large volumes of perception data that are irrelevant to the driving safety, exceeding available communication bandwidth. Moreover, most CP frameworks rely on pre-defined communication partners, making them unsuitable for dynamic traffic environments. This paper proposes a Spontaneous Risk-Aware Selective Cooperative Perception (SRA-CP) framework to address these challenges. SRA-CP introduces a decentralized protocol where connected agents continuously broadcast lightweight perception coverage summaries and initiate targeted cooperation only when risk-relevant blind zones are detected. A perceptual risk identification module enables each CV to locally assess the impact of occlusions on its driving task and determine whether cooperation is necessary. When CP is triggered, the ego vehicle selects appropriate peers based on shared perception coverage and engages in selective information exchange through a fusion module that prioritizes safety-critical content and adapts to bandwidth constraints. We evaluate SRA-CP on a public dataset against several representative baselines. Results show that SRA-CP achieves less than 1% average precision (AP) loss for safety-critical objects compared to generic CP, while using only 20% of the communication bandwidth. Moreover, it improves the perception performance by 15% over existing selective CP methods that do not incorporate risk awareness.