The actress born in Bradford, West-Yorkshire played the Gallifreyan companion to the fourth Doctor, Tom Baker, between 1978 and 1979 in The Key to Time season arc.
Tom Baker led the tributes to Tamm.
He said in a statement: "She was a darling companion and wonderfully witty and kind. I'm so sorry to hear of her death."
Barry Langford, her agent of 22 years, also paid tribute to one of his "closest friends".
He said: "She had a great zest for life.
"She was a fantastic actress - she played stage parts of such range, parts that would take your breath away.
"She could play any role, and do so wonderfully."
The sad news of Tamm’s death comes just a month after that of fellow companion actress Caroline John, who played Dr Liz Shaw opposite Jon Pertwee. Our thoughts and best wishes go to both families.
“I remember Mary Tamm’s first appearance so vividly - the ice Queen on the TARDIS. The Time Lady who thought the Doctor was HER companion. Perfectly brought to life by Mary, with such style and wit, you always thought she could have kicked the Doctor out of the time machine and got on with the adventure herself. A generation of little girls threw away the idea of being an assistant, and decided to fly the TARDIS for themselves.”
— Steven Moffat on Mary Tamm
Comments from past Doctor Who cast on Twitter:
I'm still reeling from the shock of Mary passing away!always a smiley friendly person whenever we met at cons
Shellshocked to hear Mary Tamm is gone. A funny, caring, talented, lovely and down to earth lady. My heart goes out to Marcus and Lauren
Just heard the terrible news that Mary Tamm has died. Completely shocked. Very sad. Thoughts go to Marcus & Lauren. RIP Mary.
Just heard the deeply sad news ! Mary Tamm has gone on her next awfully big adventure ! A lovely fun talented lady xxxxxx
I knew Mary for 44 years. Beautiful, highly intelligent, ambitious, multi-talented and funny, oh so funny... Darling woman RIP