<p>We define censored fractional Bernstein derivatives on the positive half-line based on the Bernstein–Riemann–Liouville fractional derivative. The censored fractional derivative turns out to be the generator of the censored decreasing subordinator <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(S^c = (S_t^c)_{t\geqslant 0}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>S</mi> <mi>c</mi> </msup> <mo>=</mo> <msub> <mrow> <mo stretchy="false">(</mo> <msubsup> <mi>S</mi> <mi>t</mi> <mi>c</mi> </msubsup> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>t</mi> <mo>⩾</mo> <mn>0</mn> </mrow> </msub> </mrow> </math></EquationSource> </InlineEquation>. We provide two constructions of the censored subordinator: (i) pathwise by removing those jumps from the decreasing subordinator <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\((x-S_t)_{t\geqslant 0}\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>-</mo> <msub> <mi>S</mi> <mi>t</mi> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>t</mi> <mo>⩾</mo> <mn>0</mn> </mrow> </msub> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(x&gt;0\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>x</mi> <mo>&gt;</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, that drive the path into negative territory. (ii) via a semigroup approach and the Hille–Yosida theorem. Then we show that the censored decreasing subordinator has only finite life-time, and we identify various probability distributions related to <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(S^c.\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>S</mi> <mi>c</mi> </msup> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation></p>