Here's an example that customizes the ObjectMapper for JSON (de)serialization. Note that for streaming purposes, different encoders/decoders are being used but the principle remains the same for their configuration.

    ExchangeStrategies strategies = ExchangeStrategies
            .builder()
            .codecs(clientDefaultCodecsConfigurer -> {
                clientDefaultCodecsConfigurer.defaultCodecs().jackson2JsonEncoder(new Jackson2JsonEncoder(new ObjectMapper(), MediaType.APPLICATION_JSON));
                clientDefaultCodecsConfigurer.defaultCodecs().jackson2JsonDecoder(new Jackson2JsonDecoder(new ObjectMapper(), MediaType.APPLICATION_JSON));

            }).build();

    WebClient webClient = WebClient.builder().exchangeStrategies(strategies).build();

A cleaner alternative to walrus03's answer:

RestClient.Builder restClientBuilder = RestClient.builder();
restClientBuilder.messageConverters(c -> {
    // Remove any existing MappingJackson2HttpMessageConverter
    c.removeIf(MappingJackson2HttpMessageConverter.class::isInstance);
    // Add MappingJackson2HttpMessageConverter
    c.add(new MappingJackson2HttpMessageConverter(newObjectMapper));
});

After quite some time spent debugging Spring WebFlux / Jackson library code, I was able to finally find a hint to solve the issue looking at the reactive WebFlux WebClient docs. There is some custom plumbing required to make WebClient use the default auto-configured Jackson ObjectMapper. The solution is to configure the default decoder used for processing Server-Sent Events when creating new instance of the WebClient. Here is the sample code:

@Component
public class MarketDataFetcher implements CommandLineRunner {

    // ...

    private final WebClient webClient;

    public MarketDataFetcher(ObjectMapper objectMapper) {
        webClient = createWebClient(objectMapper);
    }

    private WebClient createWebClient(ObjectMapper objectMapper) {
        return WebClient.builder()
                .codecs(configurer -> configurer.defaultCodecs()
                        .serverSentEventDecoder(new Jackson2JsonDecoder(objectMapper)))
                .baseUrl(BASE_URL)
                .build();
    }
}

ObjectMapper is automatically injected by Spring, so no @Autowired annotation is needed.

It would certainly help if this could be made more explicit in the official documentation somehow. Hope this answer can be useful for someone facing similar issue!