Is there a light (SPV) version of bitcoin core (bitcoind)? I'm trying to develop an SPV Bitcoin client which makes use of a library requiring bitcoin core (a running bitcoind service).
However, from what I understand, Bitcoin Core is full node, meaning that it stores the whole blockchain.

My question is: there exists a service which is similar to bitcoind (that is it provides a JSON-RPC server), without needing the whole blockchain?
Yes, bitcoind with pruning is similar to bitcoind, though some RPCs are disabled (naturally, because of pruning).

bitcoind in pruning mode does not store the whole blockchain, and can reduce your space usage to 2GB instead of 85GB.

See the release notes here and here.  

Update: There is also a Full Block SPV wallet PR by Jonas Schnelli. Which is even lighter than pruning mode, since it only downloads blocks from the creation of the wallet.