Is it possible to have high compression but low predictability? Can you have a process that generates a binary sequence with high compression rate (low entropy) but impossible to predict next symbol?

'impossible to predict' - sequence cannot be predicted theoretically. But I was also asking about the real life example (no polynomial algorithm).
Yes. A cryptographic stream cipher can generate an incredibly long string  (e.g. ChaCha20 can generate $256 \times 2^{128}$ bits) from a single small key (e.g. 256 bits).

If you do not know the key that was used this stream is impossible to predict.