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Off-chain channel networks are one of the most promising technologies for dealing with blockchain scalability and delayed finality issues. Parties connected within such networks can send coins to each other without interacting with the blockchain. Moreover, these payments can be "routed" over the network. Thanks to this, even the parties that do not have a channel in common can perform payments between each other with the help of intermediaries. In this paper, we introduce a new notion that we call Non-Atomic Payment Splitting (NAPS) protocols that allow the intermediaries in the network to split the payments recursively into several subpayments in such a way that the payment can be successful "partially" (i.e. not all the requested amount may be transferred). This contrasts with the existing splitting techniques that are "atomic" in that they did not allow such partial payments (we compare the "atomic" and "non-atomic" approaches in the paper). We define NAPS formally and then present a protocol that we call "EthNA", that satisfies this definition. EthNA is based on very simple and efficient cryptographic tools; in particular, it does not use expensive cryptographic primitives. We implement a simple variant of EthNA in Solidity and provide some benchmarks. We also report on some experiments with routing using EthNA.