Sadly a lot of TOTP performances from the 1960s have been lost, but check out this fairly extensive list of TOTP performances from the 1960s that exist in the archive.
Dave Clark Five performing 'Bits And Pieces'
The Hollies performing 'Just One Look'
The Rolling Stones performing 'Not Fade Away'
Gene Pitney performing 'That Girl Belongs To Yesterday'
The Supremes performing 'Baby Love'
Roy Orbison performing 'Oh Pretty Woman'
The Moody Blues performing 'Go Now'
Sonny & Cher performing 'I Got You Babe'
Jonathan King performing 'Everyone's Gone To The Moon'
The Byrds performing 'All I Really Want To Do'
The Shadows performing 'Don't Make My Baby Blue'
Herman's Hermits performing 'Just A Little Bit Better'
The Beatles performing 'All You Need Is Love'
Procol Harum performing 'A Whiter Shade Of Pale'
Pink Floyd performing 'See Emily Play'
Dave Davies performing 'Death Of A Clown'
Alan Price Set performing 'The House That Jack Built'
Tom Jones performing 'I'll Never Fall In Love Again'
The Turtles performing 'She'd Rather Be With Me'
Long John Baldry performing 'Let The Heartaches Begin'
Diana Ross & The Supremes performing 'Reflections'
Engelbert Humperdinck performing 'The Last Waltz'
Dave Dee, Dozy, Beaky, Mick & Tich performing 'Zabadak!'
Cliff Richard performing 'All My Love'
The Bee Gees performing 'Massachusetts'
Lulu performing 'Love Loves To Love Love'
The Rolling Stones performing 'Let's Spend The Night Together'
Status Quo performing 'Pictures Of Matchstick Men'
Amen Corner performing 'Bend Me, Shape Me'
Alan Price Set performing 'Don't Stop The Carnival'
Brenton Wood performing 'Gimme Little Sign'
Dave Dee, Dozy, Beaky, Mick & Tich performing 'Legend Of Xanadu'
The Move performing 'Fire Brigade'
Manfred Mann performing 'Mighty Quinn'
The Foundations performing 'Back On My Feet Again'
Herman's Hermits performing 'I Can Take Or Leave Your Loving'
Don Partridge performing 'Blue Eyes'
Julie Driscoll, Brian Auger & The Trinity performing 'This Wheel's On Fire'
Manfred Mann performing 'My Name Is Jack'
Cilla Black performing 'Where Is Tomorrow'
Sandie Shaw performing 'Monsieur Dupont'
Marvin Gaye performing 'I Heard It Through The Grapevine'
Cilla Black performing 'Surround Yourself With Sorrow'
Donald Peers performing 'Please Don't Go'
Peter Sarstedt performing 'Where Do You Go To (My Lovely)'
Cliff Richard performing 'Good Times (Better Times)'
The Hollies performing 'Sorry Suzanne'
Dave Dee, Dozy, Beaky, Mick & Tich performing 'Don Juan'
The Tymes performing 'People'
Stevie Wonder performing 'I Don't Know Why (I Love You)'
The Bee Gees performing 'The First Of May'
Love Affair performing 'One Road'
Lulu performing 'Boom Bang-a-bang'
Clodagh Rodgers performing 'Goodnight Midnight'
Marmalade performing 'Ob-la-di-ob-la-da'
The Rolling Stones performing 'Honky Tonk Women'
Scaffold performing 'Lily The Pink'