Não. Consulte Visão geral do IBM SSL
The SSL client sends a "client hello" message that lists cryptographic information such as the SSL version and, in the client's order of preference, the CipherSuites supported by the client. The message also contains a random byte string that is used in subsequent computations. The SSL protocol allows for the "client hello" to include the data compression methods supported by the client, but current SSL implementations do not usually include this provision.
The SSL server responds with a "server hello" message that contains the CipherSuite chosen by the server from the list provided by the SSL client, the session ID and another random byte string. The SSL server also sends its digital certificate. If the server requires a digital certificate for client authentication, the server sends a "client certificate request" that includes a list of the types of certificates supported and the Distinguished Names of acceptable Certification Authorities (CAs).
The SSL client verifies the digital signature on the SSL server's digital certificate and checks that the CipherSuite chosen by the server is acceptable.
…
O resumo da Microsoft é semelhante. O handshake TLS também é semelhante a esse respeito.
Na etapa 2, não parece haver uma maneira de o cliente dizer "não se preocupe em enviar um certificado de servidor, usarei meu cache".
Observe que existem vários tipos de certificados, cliente, servidor e CA. Algumas delas são armazenadas em cache.